ANGLES - Picnic Point High School · 2020-04-02 · Naming Angles Naming angles allows you to explain which angle or part of a shape you're dealing with. The first column introduces

ANGLES

ALTITUDE FREEFALL MATHEMATICS

Naming Angles Naming angles allows you to explain which angle or part of a shape you're dealing with. The first column introduces intervals, lines and rays. With intervals and lines you can name them in either direction. The interval below could be WQ or QW. You are asked to give both the names in these questions. With rays, name them in the direction of the arrow only. The ray below is called HG only, not GH. In column 2 you are asked to name the rays and also the vertex, this is the meeting point of the two rays, the ‘elbow’ if you like. The vertex is named by the letter at the point alone. To show it is a vertex we put a hat on it, like an upside-down ‘v’. To name an angle we generally use 3 letters, moving from ray-to vertex-to ray. The vertex is always in the middle. To show that it is an angle you put an angle sign in front of it or write the letters placing a hat over the vertex. When you name a shape (in column 3) you start at any point then go around the shape the one way. If the letters are in alphabetical order you might like to start at the earliest letter. No symbol is put before the name of a shape, though sometimes this is done with triangles. (e.g. ABC)

W Q

H G

ABC < ABC ^

or,

Naming Angles

Give the 2 possible names for these intervals.

1 A B

or

2 K Z

or

3 or L

D

4 or W

U

Give the 2 possible names for these lines.

5 or T Q

6 or C

S

7 or H I

8 E

V R

K i)

ii)

or i) or ii)

Name these rays

9 X F

10 N

Y

11 M

P

12

H

J

20

V

H

R

T

K

21

F

C B

D

E

A

22 E

P K

J

T

N

19

D

A B

C

Name these shapes using their letters. Then name the angle with the symbol.

Place the given symbol in the following angles

23

IHG <

FED <

BCD <

AEI <

ABC <

AED <

E

H I

G

F

A

D

C

B Note E is the point of contact

16 <

X

A

B

17

T

L Z

F

18

O M

K

Y

A

Name these angles, with the vertex as the middle letter. Use either ^ or <

< ^

Name the 2 rays and the vertex for these angles

13 K V

Y

ray

^

vertex ray

14 T

A

P

ray

vertex

ray

15 S

A L

Z

E

ray

vertex ray

ray

vertex ray

© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE

Naming Angles

Give the 2 possible names for these intervals.

1 A B

AB or BA

2 K Z

KZ or ZK

3 DL or LD L

D

4 UW or WU W

U

Give the 2 possible names for these lines.

5 TQ or QT T Q

6 CS or SC C

S

7 HI or IH H I

8 E

V R

K i)

ii)

RK or KR i) EV or VE ii)

Name these rays

9 XF X F

10 NY N

Y

11 MP M

P

12 HJ

H

J

20

HRV <

TKVRH

V

H

R

T

K

21

CDE <

ABC <

ABCDEF F

C B

D

E

A

22 E

P K

J

T

N

JEN <

JKN <

TNK <

ENTKPJ

19

DAB <

ABCD D

A B

C

Name these shapes using their letters. Then name the angle with the symbol.

Place the given symbol in the following angles

23

IHG <

FED <

BCD <

AEI <

ABC <

AED <

E

H I

G

F

A

D

C

B Note E is the point of contact

16 AXB <

X

A

B

17

T

L Z

F TZL <

FZT <

18

O M

K

Y

A

AOK <

KOM <

AOY <

Name these angles, with the vertex as the middle letter. Use either ^ or <

< ^

Name the 2 rays and the vertex for these angles

13 K V

Y

KVray

K ^

vertex

KYray

14 T

A

P ATray

A ^

vertex

APray

15 S

A L

Z

E

EAray

E ^

vertex

ESray

EAray

E ^

vertex

EZray


Classifying Angles Classification means to group into categories. There are six classifications in all:

acute angles are between 0° and 90° a right angle is exactly 90° obtuse angles are between 90° and 180° a straight angle is exactly 180° reflex angles are between 180° and 360° and a revolution is 360°

How do you remember all these? You should be able to remember the straight angle, right angle and revolution, it is the others you could mix up. Remember that as the angle increases you move through the alphabet. A (acute) is before O(Obtuse) which is before R(reflex). There is a guide at the top of the page to refer to. With the angles, the measured side is the side marked with the arc (part circle) or the right angle symbol. When asked to name angles remember that you can use either the hat or the angle sign method, as below. In column 3 the word internal is used. What does internal mean? It means the inside (interior), so the internal angles are the angles on the inside of a shape.

ABC < ABC ^

or,

Classifying Angles

Revolution360°

Right 90°

Straight 180°

Acute < 90°

Obtuse between 90° and 180°

Reflex between 180° and 360°

Classify the following angles

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

18

19

17

Complete the table below

Classification Angle

115°

333°

73°

165°

180°

200° 360°

20

Using the ray AB in all angles name an acute, obtuse and straight angle

22

acute

obtuse

straight

Q

G

B X

A

23

acute

obtuse

straight

E

I D

A

B

S

21

acute

obtuse

straight

A C B

E D

iv) reflex angles that can be

made

iii) pieces eaten to make the

largest obtuse angle

ii) pieces eaten to make a right

angle

i) acute angles that can be

made

26 A family size pizza has the first piece eaten forming an acute angle. Each piece is the same size and when removed increases the size of the angle. Find the number of:

exposed tray forms an acute angle

For the two shapes below name internal angles that are acute, obtuse or reflex.

25

acute

obtuse

reflex

Y

C

O

V

K

L

24 D

J

T

E

X

acute

obtuse

reflex


Classifying Angles

Revolution360°

Right 90°

Straight 180°

Acute < 90°

Obtuse between 90° and 180°

Reflex between 180° and 360°

Classify the following angles

1 acute

2 obtuse

3 revolution

4 right

5 reflex

6 straight

7 acute

8 reflex

9 revolution

10 obtuse

11 right

12 obtuse

13 acute

14 acute

15 reflex

16 obtuse

18 obtuse

19 reflex

17 straight

Complete the table below

Classification Angle

obtuse 115°

reflex 333°

acute 73°

obtuse 165°

straight 180°

reflex 200° revolution 360°

20

Using the ray AB in all angles name an acute, obtuse and straight angle

22

ABX <

acute

ABQ <

obtuse

ABG <

straight

Q

G

B X

A

23

ABD <

acute

ABS <

obtuse

ABE <

straight

E

I D

A

B

S

21

ABE <

acute

ABD <

obtuse

ABC <

straight

A C B

E D

iv) reflex angles that can be

made 7

iii) pieces eaten to make the

largest obtuse angle 7

ii) pieces eaten to make a right

angle 4

i) acute angles that can be

made 3

26 A family size pizza has the first piece eaten forming an acute angle. Each piece is the same size and when removed increases the size of the angle. Find the number of:

exposed tray forms an acute angle

For the two shapes below name internal angles that are acute, obtuse or reflex.

D

J

T

E

X

25

KLC <

acute

LCO <

obtuse

LKV <

reflex

Y

C

O

V

K

L

24

XET <

acute

DJX <

obtuse

ETD <

reflex


Estimating and Measuring Angles An angle is used to measure rotation, a protractor is required for this worksheet. To answer this sheet look at each angle and estimate its size, estimate meaning a skilled guess, so it is unlikely to guess the exact answer, but within say 15° would be very good. Don't use a protractor until you have estimated all of the angles, use the guide at the top right of the sheet if you need the help. When using a standard protractor ensure you use the correct scale (inside or outside, make sure the scale used starts at 0°) and if the angle is a reflex (greater than 180°), measure the outside of the angle then subtract the angle from 360°. If an angle is measured and it is above 90°, then it must be larger than a right angle, if it isn’t you have made a mistake. Use your estimation as a guide as well, and then see how close you were.

Estimating and Measuring Angles

Estimate the size of all the shaded angles and place your answers in the table. Then using a protractor measure all the angles. See how well you estimated. (Within 15° above or below)

90° 180° 270° 360°

15

1

2

3

4

5

6

7 8

9

10

11

12 13

14

16 17

18

1 7 13

No Estimate Measured No Estimate Measured No Estimate Measured

3 9 15

2 8 14

4 10 16

5 11 17

6 12 18


Estimating and Measuring Angles

Estimate the size of all the shaded angles and place your answers in the table. Then using a protractor measure all the angles. See how well you estimated. (Within 15° above or below)

90° 180° 270° 360°

1 5 ~ 25° 10° 7 300 ~ 330° 316° 13 115 ~ 140° 127°

No Estimate Measured No Estimate Measured No Estimate Measured

3 85 ~ 95° 90° 9 95 ~ 125° 110° 15 270 ~ 300° 287°

2 125 ~ 155° 141° 8 180° 180° 14 20 ~ 35° 27°

4 20 ~ 40° 32° 10 65 ~ 85° 75° 16 110 ~ 135° 123°

5 190 ~ 220° 206° 11 265 ~ 275° 270° 17 150 ~ 170° 166°

6 40 ~ 60° 50° 12 190 ~ 220° 203° 18 300 ~ 330° 316°

15

1

2

3

4

5

6

7 8

9

10

11

12 13

14

16 17

18


Constructing Angles The previous sheet involved measuring angles, this sheet is about the construction of angles. The method is as follows:

Draw a horizontal line Measure the given angle from the end of the drawn line, plotting a point at the angle Draw a line from the end of the line to the point Label the angle with the letters supplied in the question, remember the vertex is point on the elbow of the angle. Draw an arc or sector at the vertex to show the angle is the inside or outside of the angle. If the angle is greater than 180° then calculate (360° - the angle) the obtuse/acute angle and use that angle. Draw the sector on the other side of the angle.

A

B C

A completed angle

A

B C

A completed angle with sector

A

C B

A completed reflex angle with sector

Vertex

Constructing Angles

Draw these acute angles

1 Construct a 30° angle ABC <

2 Construct a 60° angle EVT <

3 Construct a 37° angle NTY <

4 Construct an 73° angle ALR <

Draw these obtuse angles

5 Construct a 110° angle FZT <

6 Construct a 165° angle TRE <

7 Construct a 138° angle JPG <

8 Construct an 97° angle NJD <

Draw these reflex angles

9 Construct a 240° angle BYC <

10 Construct a 190° angle XOJ <

11 Construct a 337° angle JVA <

12 Construct an 254° angle PUK <


Constructing Angles

Draw these acute angles

1 Construct a 30° angle ABC <

2 Construct a 60° angle EVT <

3 Construct a 90° angle NTY <

4 Construct an 73° angle ALR <

Draw these obtuse angles

5 Construct a 110° angle FZT <

6 Construct a 165° angle TRE <

7 Construct a 138° angle JPG <

8 Construct an 97° angle NJD <

Draw these reflex angles

9 Construct a 240° angle BYC <

10 Construct a 190° angle XOJ <

11 Construct a 337° angle JVA <

12 Construct an 254° angle PUK <

N

T Y

A

L R

A

B C

E

V T

F

Z T

T

R E

J

P G

N

J D

Y B

C

J X O

J

A

V

K U

P


Creating an Isometric Cube Isometric drawings are done using 30° angles. These are either drawn with a protractor or a 30°/60° set square. Make sure you don’t use a 45° set square, as then you will be drawing an oblique drawing. Method:

From the point on the worksheet, or from a point on your page draw a 30° line to the left and another to the right. Measure off 10 cm and draw solid lines

Draw 3 vertical lines: from the start point and from ends of both lines you have just drawn. Make sure that these lines are vertical by measuring the distance from the side border of the sheet, make sure it is the same distance at the top and the bottom.

Join the 3 ends together

Draw a light construction line straight up the centre line, and construct the 30° angled lines from each end. Join up and you are done!

Time Out Activity - Using Angles to Create an Isometric Cube

Construct an isometric cube with sides 10 cm using your protractor skills and a ruler. Hint: draw the bottom edges first then the 3 vertical lines.

A smaller version of the finished product

30° 30°


Time Out Activity - Using Angles to Create an Isometric Cube

Construct an isometric cube with sides 10 cm using your protractor skills and a ruler. Hint: draw the bottom edges first then the 3 vertical lines.

A smaller version of the finished product

30° 30°


Adjacent Angles Adjacent means next to, in the case of angles this means that the angle shares a common ray (or arm) and vertex with the angle beside it. In Column 1 the table requires you to add the two angles together to get a total, the sum of two adjacent angles. This continues down the column only with graphical representation, note that these angles aren't to scale, so don’t use the angles as a guide. In Q.3 there are three angles but only two angles have numbers, the third is a right angle which is 90°. It is suggested that you write in the '90°', that way you won't forget it in the addition. In the 2nd Column Q.6 asks you to name the adjacent angle. This requires you to name the two angles that touch the given angle. As the layout is circular that means each angle has an adjacent angle on each side of it, don't name one angle then the same angle with the letters reversed. Q. 7 is another table which this time gives you the total of two angles, and one of the angles. Subtract the angle from the total and you have the answer. The same method is used for the rest of the sheet Subtract the known angles from the total to obtain the unknown angle. Except Q.10 & 12, with these use division! Q.10 gives you the hint.

Adjacent Angles

Angles 1 and 2 are adjacent find the sum of the two angles

Angle 1

35°

19°

7°

115°

21°

273°

311°

Angle 2

42°

27°

96°

153°

117°

24°

46°

Total

77°

1

Find the size of the angle formed by adding these adjacent angles.

2

93° 34°

x

x = +

x =

4 A

T

I

N

ANI < = 77°

TNI < = 68°

TNA = <

3 x

68° 27°

x =

x =

5

41°

65° 29°

k

Name two adjacent angles to the following

H

J U A

F C

B

6

ABC < : &

HBF < : &

UBC < : &

This time subtract. Given the total and one angle, find the other angle.

Total

38°

52°

191°

311°

187°

304°

339°

Angle 1

15°

208°

11°

221°

Angle 2

17°

115°

147°

7

8

m

162°

m = -

m = 9

149°

t 84°

You are given the total now subtract or divide to find the missing angles.

Keep going!

10 e 64°

e e =

2e =

13 119°

c

14

d 115°

265°

11

240°

85°

115°

y

15

q

255°

12

t t t

87°


Adjacent Angles

Keep going!

12

t t t

87° 29° = t

87° 3t =

10 e 64°

e 32° e =

64° 2e =

13 119°

c 90°

29° = c

119° - 90° c =

14

d 115° 90°

265°

60° = d

265° - 115° - 90° d =

11

240°

85°

115°

y

40° = y

240° - 115° - 85° y =

15

q 90°

255° 90°

75° = q

255° - 90° - 90° q =

Angles 1 and 2 are adjacent find the sum of the two angles

Angle 1

35°

19°

7°

115°

21°

273°

311°

Angle 2

42°

27°

96°

153°

117°

24°

46°

Total

77°

46°

103°

268°

138°

297°

357°

1

Find the size of the angle formed by adding these adjacent angles.

2

93° 34°

x

93° x = + 34°

127° x =

4 A

T

I

N 77°

68°

ANI < = 77°

TNI < = 68°

145° TNA < =

TNA < 68° + 77° =

3 x

68° 27° 90°

185° = x

27° + 90° + 68° x =

5

41° 90°

65° 29°

k

225° = k

41° + 90° + 65° + 29° k =

Name two adjacent angles to the following

H

J U A

F C

B

6

ABC < : & ABU < CBH <

HBF < : & FBJ < HBC <

UBC < : & UBJ < CBH <

This time subtract. Given the total and one angle, find the other angle.

Total

38°

52°

191°

311°

187°

304°

339°

Angle 1

15°

35°

76°

208°

11°

221°

192°

Angle 2

23°

17°

115°

103°

176°

83°

147°

7

8

m

162°

90°

162° m = - 90°

72° m = 9

149°

t 84°

65° = t

149° - 84° t =

You are given the total now subtract or divide to find the missing angles.


Complementary and Supplementary Angles Complementary angles are angles that add to 90° and supplementary angles are angles that add to 180°. How do you remember which is which? C comes before S in the alphabet, as 90° comes before 180°. Or look at the construction below. In Column 1 the first three questions ask you to verify that the two angles are complementary. Add the two angles together, if the sum equals 90° then they are. For the rest of Column 1 you have to find the angle that when added to the angle given, has a total of 90°. You answer this by subtracting the given angle from 90°. The most common mistake is thinking that a right angle is 100°, or at least using it in calculations, remember 90° not 100°! The 2nd Column has the same layout as the first but instead of the angles adding to give 90° they should total 180°, to be supplementary. The remainder of the column requires you to subtract the given angle from 180° to get the answer. When a right angle is involved write in 90°, this process should always be done as the angle may be overlooked. The 3rd column is a mixture of the first 2 columns.

C S The C can be changed to a 9 (90°) and the S can be changed to an 8 to remember 180°

Complementary and Supplementary Angles

Are the following angles complementary?

1

55°

45°

Circle: Yes / No

2

68°

22°

Circle: Yes / No

3 42°

48°

Circle: Yes / No

Find the unknown adjacent complementary angle

4

= d

d = 90° -

d 32°

5

t 56°

6

y

17°

7

w 41°

8 s

25°

9 h 72°

Are the following angles supplementary?

10 24° 156°

Circle: Yes / No

11 127°

63°

Circle: Yes / No

12 87° 83°

Circle: Yes / No

Find the unknown adjacent supplementary angle

13

= k

k = 180° -

k 46°

14

x 123°

15 u

22°

16 w

17 p

136°

18

c

169°

19

b

33°

Find the missing angle

20 y

17°

21

j 141°

22 m

37°

23 p

102°

24 d 61°

25 y

9°

26 n

39°

27 a

50°

28 h

29

e

152°

30

k 43°


Complementary and Supplementary Angles

Find the unknown adjacent complementary angle

4

58° = d

d = 90° - 32°

d 32°

8

65° = s

90° - 25° s = s

25°

5

34° = t

90° - 56° t = t 56°

Are the following angles complementary?

1

55°

45°

Circle: Yes / No

2

68°

22°

Circle: Yes / No

3 42°

48°

Circle: Yes / No

6

73° = y

90° - 17° y =

y

17°

7

49° = w

90° - 41° w = w 41°

9

18° = h

90° - 72° h = h 72°

Are the following angles supplementary?

10 24° 156°

Circle: Yes / No

11 127°

63°

Circle: Yes / No

12 87° 83°

Circle: Yes / No

Find the unknown adjacent supplementary angle

13

134° = k

k = 180° - 46°

k 46°

14

57° = x

x = 180° - 123°

x 123°

15

158° = u

u = 180° - 22° u

22°

17

44° = p

p = 180° - 136° p

136°

18

11° = c

c = 180° - 169°

c

169°

19

147° = b

b = 180° - 33°

b

33°

16

90° = w

w = 180° - 90° w 90°

20

163° = y

y = 180° -17° y

17°

21

39° = j

j = 180° - 141°

j 141°

23

78° = p

p = 180° - 102° p

102°

26

141° = n

n = 180° - 139° n

39°

29

28° = e

e = 180° - 152°

e

152°

28

90° = h

h = 180° - 90° h

90°

22

53° = m

m = 90° - 37° m

37°

24

29° = d

d = 90° - 61° d 61°

25

81° = y

y = 90° - 9° y

9°

27

40° = a

a = 90° - 50° a

50°

30

47° = k

k = 90° - 43°

k 43°

Find the missing angle


Complementary Angles Complementary angles are angles that add to 90°. Supplementary angles add to 180° (the next sheet). How do you remember which is which? C comes before S in the alphabet, as 90° comes before 180°. Or look at the construction below. In Column 1 you are asked to find the complement, this means find the angle that when added to the angle given, has a total of 90°. You do this by subtracting the given angle from 90°. This continues down the column, subtract the angle given on the diagram from 90° to find the answer. The most common mistake is thinking that a right angle is 100°, or at least using it in calculations remember 90° not 100°! The 2nd Column through to Q 16 in the 3rd Column is an extension of this. The same method is used only more than one angle is subtracted from 90°. Question 7 shows the method of working required. Questions 17 and 18 require you to add the letters together, these equal 90°. Then divide by the number in front of the letter to get the value of the pronumeral. Note that the value of the pronumeral is being found, this isn’t necessarily the value of the angle. Questions 19 and 20 are more difficult questions and require an extra step. The letters are added on the left of the equals sign, and the subtraction of the angle from 90° is put on the right side. See the example below.


Example

45° = 5d

90° - 45° 5d =

9° = d Divide by

5 4d

d 45°

Complementary Angles

Find the complementary angle to those given in the table below.

Angle

30°

45°

73°

21°

9°

57°

84°

Complement

60°

1

Find the complementary angle for the following

3

d 53°

2

x 27° x =

90° x = -

4 h 42°

5 e 78°

6

b 36°

Now there are more than 2 angles. Solve these.

7

a = 90° - 33° - 33°

a 33°

33°

8

y

24° 22°

13

n

21° 27°

16°

9

t

13°

11°

10

m 54° 18°

11

k

75°

8°

12

c 31°

37°

14

w 8° 5°

17°

Use division to find the value of the letters.

17

a a

a

19

e e

40°

15

g

17°

48°

9°

16

j

34°

25° 14°

18 3d 3d

20

3k k

42°


Complementary Angles

Find the complementary angle to those given in the table below.

Angle

30°

45°

73°

21°

9°

57°

84°

Complement

60°

45°

17°

69°

81°

33°

6°

1

Find the complementary angle for the following

3

37° = d

90° - 53° d = d 53°

4 h 42°

48° = h

90° - 42° h =

5

12° = e

90° - 78° e = e 78°

6

54° = b

90° - 36° b =

b 36°

2

x 27° 63° = x

90° x = - 27°


7

24° = a

90° - 33° - 33° a =

a 33°

33°

8

44° = y

90° - 24° - 22° y =

y

24° 22°

9

66° = t

90° - 11° - 13° t = t

13°

11°

10

18° = m

90° - 54° - 18° m = m 54° 18°

11

7° = k

90° - 8° - 75° k = k

75°

8°

12

22° = c

90° - 37° - 31° c = c 31°

37°

13

26° = n

90° - 21° - 27° - 16° n =

n

21° 27°

16°

14

60° = w

90° - 17° - 8° - 5° w =

w 8° 5°

17°

15

16° = g

90° - 9° - 17° - 48° g =

g

17°

48°

9°

16

17° = j

90° - 34° - 25° - 14° j =

j

34°

25° 14°

Use division to find the value of these unknowns.

17

30° = a

90° 3a =

a a

a

15° = d

90° 6d = 3d 3d

18

19

e e

40° 50° = 2e

90° - 40° 2e =

25° = e 20

3k k

42° 48° = 4k

90° - 42° 4k =

12° = k


Supplementary Angles Supplementary angles are angles that add to 180°. How do you remember the difference between supplementary and complementary? C comes before S in the alphabet, just as 90° comes before 180°. Or look at the construction below. In Column 1 you are asked to find the supplement, this means find the angle that when added to the angle given, has a total of 180°. You do this by subtracting the given angle from 180°. This continues down the column, subtract the angle given on the diagram from 180° to get the answer. The 2nd Column has a table to complete, this time both the complementary and supplementary angles are required. Subtract the angle from 180° to get the supplement and from 90° for the complement. Q 9 through 16 add more angles into the problem, but the problem is still solved the same way, by subtracting all the given angles from 180°. Questions 17 and 19 require you to add the letters together, these equal 180°. Then divide by the number in front of the letter to get the value of the pronumeral. Note that the value of the pronumeral is being found which isn’t necessarily the value of the angle (see Question 20). Question 18 is a more difficult question and requires an extra step. The letters are added on the left of the equals sign, and the subtraction of the angle from 90° is put on the right side. See the example below.


Example

35° = 5d

180° - 145° 5d =

7° = d Divide by

5 4d

d

145°

Supplementary Angles

Find the supplementary angle to those given in the table below.

1 Angle

110°

45°

162°

16°

123°

27°

177°

Supplement

70°

Find the supplementary angle for the following

2 x 50°

= x

180° x = -

t 22°

3

4 c

117°

5

a 164°

7

p

6 u

38°

8

Angle

25°

65°

13°

79°

57°

3°

41°

Supplement

Complement

For the angles below find both the supplement and the complement.


9

l 55°

10

w 40°

11

d 63° 20°

12 40°

m 106°


13

b 35° 45°

15°

17

g g g

20 With the value of x found above, find the value of:

= 2x = 4x

18

20°

4e

14 a

25° 42°

15 h

21° 120° 25°

16

k 15°

24°

19

4x x 2x

2x


Supplementary Angles

Find the supplementary angle to those given in the table below.

1 Angle

110°

45°

162°

16°

123°

27°

177°

Supplement

70°

135°

18°

164°

57°

153°

3°

Find the supplementary angle for the following

2 x 50°

130° = x

180° x = - 50°

t 22°

158° = t

180° - 22° t =

3

4

63° = c

180° - 117° c = c 117°

5

16° = a

180° - 164° a =

a 164°

6

142° = u

180° - 38° u = u 38°

7

90° = p

180° - 90° p = p

8

Angle

25°

65°

13°

79°

57°

3°

41°

Supplement

155°

115°

167°

101°

123°

177°

139°

Complement

65°

25°

77°

11°

33°

87°

49°

For the angles below find both the supplement and the complement.


9

35° = l

180° - 90° - 55° l =

l 55° 90°

10

50° = w

180° - 90° - 40° w =

w 40° 90°

11

97° = d

180° - 63° - 20° d =

d 63° 20°

12

34° = m

180° - 106° - 40° m =

40° m 106°


13

85° = b

180° - 15° - 45° - 35° b =

b 35° 45°

15°

14 a

25° 90° 42°

23° = a

180° - 90° - 42° - 25° a =

15

14° = h

180° - 120° - 21° - 25° h =

h 21° 120° 25°

16

51° = k

180° - 15° - 90° - 24° k =

k 15° 90°

24°

17

g g g

180° = 3g

60° = g

18

20°

4e 160° = 4e

180° - 20° 4e =

40° = e

19 180° = 9x

20° = x 4x x 2x

2x

20 With the value of x found above, find the value of:

40° = 2x 80° = 4x


Angles at a Point Just like a circle, the angle sum at a point is 360°. To find the size of an unknown angle subtract all the given angles from 360°. In Column 1 the unknown angle is given a letter, just write the letter then '=' and subtract all the given angles from 360°. Some angles are right angles, write '90°' in these so that you don't forget to include the right angle in your calculation. Column 2 requires you to use your algebra skills, add the letters and then divide 360° by the number in front of the letter. Questions 13 through 15 have an additional step, use the same method to find the value of the letter, but then multiply that value by the number in front of the letter for each angle. For example if you find x = 25° and the angle in the question is 3x, then 3x = 3 × 25° = 75°. If the question asks 'to solve for x', then you don't need to do this, it is only when you are asked for the angle, as the angle is 3x you must find the size of the angle, as the size of x isn't sufficient. Column 3 is a harder column and some students may experience difficulty with these. The steps are a combination of the earlier problems.

add the letters together and write it then an equals, (in question 16. e + e = 2e) then subtract the given angles (remember the right angles) from 360° the next line write the letters sum again only evaluate 360° - the angle then use division to solve like earlier problems the last two questions are harder, they require multiplication to get the angle (like Q13-15)

Angles at a Point

Use subtraction from 360° to find the unknown angle

1

d

305° = d

d = 360° -

2 x

165°

3 y

123° 117°

4 f 109°

5 g 135°

145° 60°

6 m

189°

56°

7 b

152° 48°

65°

This time use division

8

k k k k =

3k =

9

y y y

y y

Use the same method but this time find the value of the letter and the angles

12

2k 3k k k =

6k =

2k = 3k =

10 x x x x

x x

11

a a a a

a

a a a

13

2d 3d 3d

14 3k

4k 5k

Use subtraction from 360° then division to find the unknown angles

15

280°

e e = 2e

= e

2e = 360° -

16

200°

q q

Now the angles are different

20

285°

2c 3c = 5c

5c =

2c = 3c =

17

120°

a a a

19

156°

n n n

n

21

2d

3d


Angles at a Point

This time use division

10 x x x x

x x

60° x =

360° 6x =

11

45° a =

360° 8a =

a a a a

a

a a a

8

k k k 120° k =

360° 3k =

9

72° y =

360° 5y = y y y

y y

Use the same method but this time find the value of the letter and the angles

12

2k 3k k 60° k =

360° 6k =

120° 2k = 180° 3k =

13

45° d =

360° 8d =

90° 2d = 135° 3d =

2d 3d 3d

14

30° = k

360° 12k = 3k 4k 5k

90° 3k = 120° 4k =

150° 5k =

Use subtraction from 360° then division to find the unknown angles

15

280°

e e 80° = 2e

40° = e

2e = 360° - 280°

16

160° = 2q

80° = q

2q = 360°- 200°

200°

q q

17

240° = 3a

80° = a

3a = 360° - 120°

120°

a a a

18

204° = 4n

51° = n

4n = 360° - 156° 156°

n n n

n

Now the angles are different

19

285°

2c 3c 75° = 5c

15° = c

5c = 360° - 285°

30° 2c = 45° 3c =

20

270° = 5d

54° = d

5d = 360° - 90°

108° 2d = 162° 3d =

90° 2d

3d

7

95° = b

360° - 152° - 48° - 65° b =

b 152°

48° 65°

Use subtraction from 360° to find the unknown angle

1

d

305° 55° = d

d = 360° - 305°

2 x

90°

165°

105° = x

360° - 90° - 165° x =

3 y

123° 117°

120° = y

360° - 123° - 117° y =

5

20° = g

360° - 135° - 145° - 60° g =

g 135° 145° 60°

6

25° = m

360° - 189° - 90° - 56° m =

m

189°

90° 56°

4

71° = f

360° - 109° - 90° - 90° f =

f 109° 90°

90°


Vertically Opposite Angles Vertically opposite angles are equal when formed by intersecting straight lines. The diagram below illustrates this. Don't let the word 'vertical' confuse you, as you can see if angles are opposite each other horizontally they are also considered vertically opposite (equal). In Column 1 identify the vertically opposite angle, by drawing a star in the angle and then also by naming it. Write the name of the angle on the line and colour the star beside it the same colour as the one you placed on the diagram. Q 5 is more difficult as many angles are all on the one diagram so take care with this question. Column 2 asks you to find the value of the letter, this is just identification, (the answer is in front of you) for Q. 6 through 9. But from Q 10 on…. some mathematics is required, this will always be subtraction or division look at the examples below. With column 3 Questions 15 and 16 use the same method as the example above them, establishing two angles from identification then the 3rd by supplementary angle methods, look at the example at the top of the column. The last two questions are more difficult and require you to choose vertically opposite or supplementary methods to solve them.

180°

62°

a 30°

a = 62° - 30° a = 32°

Example 2

62°

a a

2a = 62° a = 31° Divide by 2

Example 3 Example 1

62°

This angle is vertically

opposite = 62°

This angle is the supplement 180°

- 62° = 118°

a = 62° - 30° a = 32°

62° Find total

first

Example 4

42°

a 30°

20°

Vertically Opposite Angles

Name the vertically opposite angle to the following angles

Name and label with a star the vertically opposite angle to the one marked with a star

5 A

D B

E

F

C

H

G

I

vertically

opposite IBG <

HBD <

ABC <

CBD <

4

A

P

B

T

C

X

Y

D

O

1 C D

B A E

Colour this star the same as the one you put in the angle

<

3 H U

X L

A

F

N

2 K T

Y S G

Find the value of the letters below

6 m

82° = m

7

k 135°

14

c

56° 26° 39°

8

x 71° 57°

e

9

w 67°

23°

n

10

54° 2p

11 r

82°

33°

12 n

122°

13 v 38° v

Use vertically opposite and supplementary angle properties to find the value of these letters

Example

62° = x

180° - 41° - 77° x =

41° = g

77° = u u 41°

77° g x

15

a

70°

20°

c

b

16 i 11°

67° d

y

17

55° x

85° b

18 k 48°

114° q


Vertically Opposite Angles

Find the value of the letters below

6 m

82° 82° = m

7

k 135° 135° = k

8

x 71° 57°

e

71° = e

57° = x

9 23° = n

67° = w w 67°

23°

n

10 54° = 2p

27° = p 54°

2p

11 r

82°

33°

49° = r

82° - 33° r =

12 32° = n

122° - 90° n = n

122° 90°

14

9° = c

65° - 56° c = c

56° 26° 39°

65°

13 v 38° v

38° = 2v

19° = v

Use vertically opposite and supplementary angle properties to find the value of these letters

Example

62° = x

180° - 41° - 77° x =

41° = g

77° = u u 41°

77° g x

15

90° = b

180° - 70° - 20° b =

70° = a

20° = c

a

70°

20°

c

b

16

102° = y

180° - 11° - 67° y =

11° = d

67° = i

i 11°

67° d

y

17

55° x

85° b

30° = x

85° - 55 x =

95° = b

180° - 55° - 30° b =

18 k 48°

114° q

18° = k

k = 180° - 114° - 48°

66° = q

48° + 18° q =

Name the vertically opposite angle to the following angles

Name and label with a star the vertically opposite angle to the one marked with a star

5 A

D B

E

F

C

H

G

I

vertically

opposite CBE < IBG <

ABF < HBD <

GBH < ABC <

FBG < CBD <

4

A

P

B

T

C

X

Y

D

O COA <

1 C D

B A E

Colour this star the same as the one you put in the angle

CEB <

3 H U

X L

A

F

N

ANH <

2 K T

Y S G

TGK <


Parallel Lines and the Transversal When parallel lines are crossed by a line the angles formed are related to each other by properties. This will require you to remember certain words and their meaning. The first word is 'transversal' this is the name of the line that cuts across parallel lines. In column 1 you are asked to name the transversal and the parallel lines, write the 2 letters that represent the line, this can be done either way as the example below shows. The naming of angles uses the 3 point method, there is 4 possible answers for some questions as the example below shows, realise that if the answer doesn't match your answer you still may be correct, but the middle letter (vertex) must always be the same. When lines cross, pairs of angles are made. Co-interior angles (called C angles) are the pair of angles formed on one side of the transversal inside the parallel lines, a 'c' can be formed around the angles (one is back to front). There are only two co-interior angle pairs in a standard 2-line 1 transversal problem. Remember the word interior means on the inside (of the parallel lines and the transversal). Alternate angles (called Z angles) are also on the inside of the parallel lines but unlike co-interior angles they are on opposite sides of the transversal. A 'Z' can be made by the lines that include these angles. Both alternate and co-interior angles are inside the two parallel lines. Corresponding angles (called F angles) require one point to be on the inside of the parallel lines and the other to be on the outside. An 'F ' or back to front ‘F’ is made by the lines that include these angles. With corresponding angles the angle is in exactly the same position on both points of intersection. Column 3 asks you to write either 'alternate', 'co-interior' or 'corresponding'. Questions 32 through 37 are more challenging but the same method is used.

F

Y

D

B

T

V A

M

VAM or VAB or MAV or BAV

AMD or FMD or DMA or DMF

TMB or BMT

Lines are VY or YV and TD or DT Transversal is either FB or BF

F

Y

D

B

T

V

TRANSVERSAL

LINES

Alternate Angles Co-interior Angles Corresponding Angles

Parallel Lines and the Transversal

Using the letters name the parallel lines (L) and the transversal (T)

1

E

C

B

A

D

F

L:

L:

T:

2

H

Q

M

X

T

J

L:

L:

T:

3

D

F U

Z P

M

L:

L:

T:

4

C X

B

I K

V

L:

L:

T:

Using the assigned letters name the 2 angles with symbols

5

A

C

H

E

G

B

F

D

6

A

F

U

D

X

H

P

R

7

S

W

D

V B

E

M

T

Show the co-interior angle by colouring the circle (the numbers are for marking purposes)

8 2 1 3

5 4 7 6

9 3 2 6 7

4 1 5

13 2

3 4 5 7 6 1

12 2 7 3

4

5 6

1

10 7 6

4 1

3 2

5

11 2

5

7

3

1

6

4

Now colour the alternate angle

14 2 1 4 3

5 7 6

15 2 4 1 3

7 6 5

19 5

7 4 2 3 6 1

18 5 4 1

2 3

7

6

16 3 5

6 2

4 7

1

17 3

6

7

2

5

1

4

Now colour the corresponding angle

20 1 3 2

5 4 7 6

21 5 6 4 2

7 3 1

22 6 1 2

4

3 7

5

23 2

4

1

7

3

6

5

State if these angles are corresponding, alternate or co-interior.

24

25

26

27

28

29

30

31

36

37

34

35

33

32


Parallel Lines and the Transversal

Using the letters name the parallel lines (L) and the transversal (T)

1

E

C

B

A

D

F

CD L:

EF L:

AB T:

2

H

Q

M

X

T

J

QT L:

HJ L:

XM T:

3

D

F U

Z P

M

DP L:

FM L:

ZU T:

4

C X

B

I K

V

IC L:

KX L:

BV T:

Using the assigned letters name the 2 angles with symbols

5

DFG <

ADF <

A

C

H

E

G

B

F

D

6

XPH <

PRU <

A

F

U

D

X

H

P

R

7

VTE <

TEB <

S

W

D

V B

E

M

T

State if these angles are corresponding, alternate or co-interior.

24

corresponding

25

alternate

26

alternate

27

co-interior

28

corresponding

29

co-interior

30

corresponding

31

co-interior

36

alternate

37

corresponding

34

co-interior

35

alternate

33

corresponding

32

co-interior


Show the co-interior angle by colouring the circle (the numbers are for marking purposes)

8 2 1 3

5 4 7 6

9 3 2 6 7

4 1 5

13 2

3 4 5 7 6 1

12 2 7 3

4

5 6

1

10 7 6

4 1

3 2

5

11 2

5

7

3

1

6

4

Now colour the alternate angle

14 2 1 4 3

5 7 6

15 2 4 1 3

7 6 5

19 5

7 4 2 3 6 1

18 5 4 1

2 3

7

6

16 3 5

6 2

4 7

1

17 3

6

7

2

5

1

4

Now colour the corresponding angle

20 1 3 2

5 4 7 6

21 5 6 4 2

7 3 1

22 6 1 2

4

3 7

5

23 2

4

1

7

3

6

5

Parallel Line Angle Properties When two parallel lines are crossed by a transversal, pairs of angles are made that we can use mathematically. The previous sheet taught you alternate, co-interior and corresponding angles, now we can use their angle properties, which are:

corresponding angles are equal if lines are parallel alternate angles are equal if lines are parallel co-interior angle sum is 180° if lines are parallel (the two angles add to 180°)

In Column 1 an angle is given and an unknown angle is required. The unknown angle will be either the alternate, corresponding or co-interior angle to the given angle. The first step is identifying which. Once identified if the angle is the alternate angle or the corresponding angle it is the same as the given angle, no working required. If the relationship is co-interior then subtract the given angle from 180° to obtain the answer. Column 2 gives both angles, identify if the relationship between the 2 angles is either alternate or corresponding. If it is, the angles must be the same for the lines to be parallel, if the angles aren't equal then the lines aren't parallel. If the relationship between the 2 angles is co-interior then the sum of the two angles must equal 180° for the lines to be parallel. If they don't add to 180º then the lines aren't parallel. This includes if the angles are the same, the same co-interior angle means the lines aren't parallel, unless the angles are 90°. See the example at the top of the column. Column 3 is the same as column 1 in procedure, the only difference being that you have to divide the angle by the number in front of the letter to get your answer. The last 3 questions require a subtraction to take place before the division.

Alternate Angles Co-interior Angles Corresponding Angles

Corresponding angles are equal if the lines are parallel

Alternate angles are equal if the lines are parallel

Co-interior angles add to 180° if the lines are parallel

Parallel Lines Angle Properties

Find if the lines below are parallel, give a brief reason for your decision.

Example

Corresponding angles not equal

so lines aren't parallel.

91°

89° Circle: Yes / No Parallel?

88° 88° Circle: Yes / No Parallel? 10

98°


12

85° 85°

Circle: Yes / No Parallel?

13

93°


14

11


92° 88°

Find the value of x

Co-interior angles require an extra step, find the value of x in these

Example

60°

3x

x = 20°

3x = 60°

15

130° 2x

16 120° 4x

17

6x

18

75° 3x

19

66°

3x

20

70° 2x

2x = 180° -

2x =

21 135°

5x

22

153°

3x

State if alternate, co-interior or corresponding. Then find the value of the letter.

Example corresponding

57° = x 57°

x

1 117°

e

9

g

3

87° y

4

17°

k

5 112°

a

8

143° w

2 153°

d

6

77° m

7

c


Parallel Lines Angle Properties

Find if the lines below are parallel, give a brief reason for your decision.

Example



91°


Alternate angles equal so lines

are parallel.

88° 88° Circle: Yes / No Parallel? 10



98°


12

Co-interior angle sum isn't

180° so lines aren't parallel

85° 85°


13

Corresponding angles equal so

lines are parallel.

93°


14

11

Co-interior angle sum is 180°

so lines are parallel


92° 88°

Find the value of x

Co-interior angles require an extra step, find the value of x in these

Example

60°

3x

x = 20°

3x = 60°

15

130° 2x x = 65°

2x = 130°

16 120° 4x

x = 30°

4x = 120°

17

6x

x = 15°

6x = 90°

18

75° 3x x = 25°

3x = 75°

19

66°

3x x = 22°

3x = 66°

20

70° 2x

x = 55°

2x = 180° - 70°

2x = 110°

21 135°

5x

5x = 180° - 135°

5x = 45° x = 9°

22

153°

3x 3x = 180° - 153°

3x = 27° x = 9°

State if alternate, co-interior or corresponding. Then find the value of the letter.

Example corresponding

57° = x 57°

x

1 117°

e

alternate

e = 117°

9 90°

g corresponding

g = 90°

3

87° y alternate

y = 87° 4

17°

k

corresponding

k = 17°

5 112°

a

corresponding

a = 112°

8

143° w alternate

w = 143°

2 153°

d

co-interior

d = 27º

d = 180º - 153º

6

77° m

co-interior

m = 103º

m = 180º - 77º

7

c all 3 apply

c = 90º

c = 180º - 90º


Further Parallel Lines If two parallel lines are crossed by a transversal eight angles are formed. Given one angle, all of the other angles can be found by using these properties:

supplementary, the angles adjacent (next to it) to the angle can be found by subtracting the given angle from 180° vertically opposite, the angle opposite the given angle is the same size as the angle then parallel line properties are used. Alternate, corresponding and co-interior can be used (it isn't necessary to use them all)

In column 1 a single angle is given, you are required to find the value of the black dotted angle in two solution 'moves'. The first step requires you to use supplementary or vertically opposite methods then for the second step use one of the parallel line properties. The example below shows the angle given is 108°, vertically opposite was used to find 5 then co-interior to the black dotted angle. Note we could also use two other methods. Supplement to 4 then alternate to black dot or supplement to 6 and corresponding to black dot. Either way is correct. Note that the dots can be coloured, ensure that your colour choice is matched on the solution line. Column 3 is an extension, this time the seven other angles are required. Use any method you like just make sure you give a reason, like the example below.

Example 1

2

3

4

5

108°

6

= 108° as vertically opposite

= 72° as co-interior to

Example

= 68° as supplement

= 112° as vertically opposite

= 68° as supplement

= 68° as co-interior

= 112° as alternate

= 68° as supplement to

= 112° as corresponding

5 1 3

4

6 2 7

112°

Further Parallel Lines

Now using the same skills find all the angles on the diagram, with reasons.

10 38°

3 2

1

11

103° 3

4 5

2

1

6

7

12

5 1 3

4

6 2 7

132°

Find the vertically opposite or supplementary angle and use it to solve for the un-known angle, give a reason

1

3

2 5

4 6

1 55°

2 1

2

3

4

5

102°

6

24° 1

3 2 5

4

6

4

Example 4 3

5

1 65°

7 6

= 115° supplement to 65°

= 115° alternate to •

3

5

2

126° 3

4 7

6

1

5

88°

3

5

4 6

1

2

7

5

3

2 4

6 1

9

4

6

5 77°

2

3

1

6

1

3 2

4 148° 5 6

8

165°

2

1

4

3 6

5


Further Parallel Lines

Now using the same skills find all the angles on the diagram, with reasons.

10 38°

3 2

1

(1)= 142° as supplement

(2)=38° as vertically opposite


11

(3)= 77° as 5 co-interior


(4)= 103° as alternate


(7)= 103° as vertically opposite

(1)= 103° as corresponding

(2)= 77° as supplement to •

103° 3

4 5

2

1

6

7

12


(1)= 132° as vertically opposite


(4)= 48° as co-interior

(5)= 132° as alternate

(6)= 48° as supplement to •

(7)= 132° as corresponding

5 1 3

4

6 2 7

132°

Find the vertically opposite or supplementary angle and use it to solve for the un-known angle, give a reason

1


= 55° as 2 corresponds to •

3

2 5

4 6

1 55°

2 1

2

3

4

5

102°

6


= 78° as 5 co-interior to •

24° 1

3 2 5

4

6

4

(3) = 156° as supplement

= 156° as corresponds to •

Example 4 3

5

1 65°

7 6


(4) = 115° as 4 alternate to •

3



5

2

126° 3

4 7

6

1

5


= 88° as co-interior to •

88°

3

5

4 6

1

2

7



5

3

2 4

6 1

9



4

6

5 77°

2

3

1

6



1

3 2

4 148° 5 6

8



165°

2

1

4

3 6

5


Measuring Angles in Triangles This exercise is an introductory exercise to show that the angle sum of a triangle is 180°. Use a protractor to measure each angle in the triangles, note you should expect to experience some error in any measurement exercise, your total may be out by a degree. Follow these steps:

measure the angles with a protractor and then using the addition spaces in the bottom left corner add them and compare the total to 180° Repeat for all the triangles

Why are there errors? As the angles are not exactly a whole degree you may round them up or down, this will affect your total.

Measuring Angles in Triangles

Measure the 3 angles in each of the triangles, then add them to get 180°,allow for small

2

3

4

5

6

1

4

+

5

+

6

+

1

+

2

+

3

+


Measuring Angles in Triangles

Measure the 3 angles in each of the triangles, then add them to get 180°,allow for small 35°

72° 72°

2

34°

90° 56°

3 37°

123°

20°

4

60° 60°

60°

5

45°

90° 45°

6

74°

68°

37°

1

4

180

+ 37 123

1

20

5

180

+ 60 60

60

6

180

+ 90 45

1

45

1

179

+ 37 68

74

2

179

+ 72 72

1

35

3

180

+ 34 56

1

90


Angles in Triangles The angle sum of a triangle is 180°. This property holds true regardless of the type of triangle. Types of triangles are:

Equilateral - all sides equal length, all (internal) angles are 60° Isosceles - two sides equal length, two angles equal, note that the equal sides are opposite the equal angles Right angled triangle - has a right angle in the triangle with the other two angles being unequal Right Isosceles triangle - is a right angled triangle with the other two angles being equal (they must be 45°) Scalene triangle - all sides and all angles unequal, the words acute and obtuse can be used to further the description Acute triangle - all angles are less than 90° Obtuse triangle - one angle is greater than 90°, note that you can't have more than one obtuse angle in a triangle

Column 1 requires you to find the missing angle. Two angles are always given, subtract these from 180° to get the answer, (example at top of column). Once you have the 3 angles describe the type of triangle it is from the selection at the top of the sheet. Often the case is to use the word ‘obtuse’ in describing a triangle but not ‘acute’. So that if obtuse isn't used the triangle must be acute. This classification will only apply to scalene and isosceles triangles, right angled triangles and equilateral triangles can't contain an obtuse angle. Column 3 tests your knowledge of isosceles and equilateral triangles. With equilateral triangles you know the angle is always 60°. So if the angle is 12w then that means 12w = 60° (÷ by 12) and w = 5°. If it was 15t then 15t = 60° (÷ by 15) and t = 4°. Note that in these questions you asked to find the value of the letter, this is different than finding the angle. For example if an angle is 20k in the corner of an equilateral triangle, k = 3° but the actual angle is still 60°. With isosceles triangles the process is the same except that instead of 60° the angle will match another angle on the diagram, question 18 is more challenging.

Angles in Triangles

60° 60°

60°

Equilateral Isosceles Right Angled Scalene Right Isosceles Acute Obtuse

Find the missing angle, then classify (name the type of ) the triangle.

Example x = 180° - 62° - 28°

x = 90° 62°

28° x

Right angled triangle Type:

1

Type:

16°

32°

y

2

Type:

36° 72°

a

3

Type:

90°

b 45°

4

Type:

60°

60°

k

5

Type:

53°

39°

m

Use the same method to solve, identify the type of triangle formed. Use 'x ='

8 A triangle has angles 23° and 46°, find the other angle

Type:


Type:

10 Two identical angles in a triangle total 120°, find the other angle

Type:

Type:

6

124° 28°

q

7

Type:

66°

24°

h

Use the side lengths to help you choose a method to find the value of the letters

11

x

13

a 64°

15

2x

2x =

16

4w 84°

12

u

23°

17

3n

4c 5t

18

4v

v

These are harder

14

d 45°


Angles in Triangles

60° 60°

60°

Equilateral Isosceles Right Angled Scalene Right Isosceles Acute Obtuse

Find the missing angle, then classify (name the type of ) the triangle.

Example x = 180° - 62° - 28°

x = 90° 62°

28° x


1 y = 180° - 32° - 16°

y = 132°

Obtuse scalene triangle Type:

16°

32°

y

2 a = 180° - 72° - 36°

a = 72°

Acute Isosceles Triangle Type:

36° 72°

a

3 b = 180° - 90° - 45°

b = 45°

Right Isosceles Triangle Type:

90°

b 45°

4 k = 180° - 60° - 60°

k = 60°

Equilateral triangle Type:

60°

60°

k

5 m = 180° - 39° - 53°

m = 88°

Acute scalene triangle Type:

53°

39°

m

Use the same method to solve, identify the type of triangle formed. Use 'x ='


Obtuse scalene triangle Type:

x = 180°- 23° - 46°

x = 111°


Acute isosceles triangle Type:

x = 180°- 56° - 68°

x = 56°

10 Two identical angles in a triangle total 120°, find the other angle

Equilateral triangle Type:

x = 180°- 120°

x = 60°

6 q = 180°- 124° - 28°

q = 28°

Obtuse isosceles triangle Type:

124° 28°

q

7 h = 180°- 24° - 66°

h = 90°


66°

24°

h

Use the side lengths to help you choose a method to find the value of the letters

11

x = 60° x 60°

60°

13

a = 180° - 64° - 64°

a = 52°

a 64°

64°

15

2x x = 30°

2x = 60°

16

4w 84°

w = 21°

4w = 84°

12 u = 23°

u

23°

17

c = 15°

4c = 60°

3n

4c 5t

t = 12°

5t = 60°

n = 20°

3n = 60°

18

v = 20°

9v = 180° 4v

v 4v

These are harder

14

d = 180° - 45° - 45°

d = 90°

d 45°

45°


Isosceles Triangles Your knowledge of isosceles triangles and their properties will be tested throughout your mathematics studies, the 3 methods of finding missing angles are dealt with on this sheet. In column 1 your are given one angle and asked to find another. With a scalene triangle this is impossible because you require 2 angles to find the remaining angle, but because these are isosceles triangles you know that the angles opposite the sides with markings are equal. Column 1 is done all the same way, place the number in the empty corner then you have your 2 angles. Then subtract the 2 angles from 180°. Look at question 1, the unlabelled angle is also 48°, so the working is s = 180° - 48° - 48° or s = 180° - 2 × 48°. The most common mistake when answering these questions is you will forget the unlabelled angle and subtract 48° from 180° instead of 48° doubled. In column 2 the second type of problem is encountered, this time you are given the non-paired angle. This time 2 × (the letter) = 180° - (the given angle). An example is at the top of the column, the entire column is done using the same method. It is up to you if you write the letter in the unlabelled corner, but it is recommended. Column 3 has written problems, 17 - 19 are the same as the previous columns, you just have to decide which method is used. The last 2 questions are harder, an example is below.

Example (questions 20 and 21) A triangle has a pair of angles which are seven times the size of the other angle, find the size of all the angles. Let x = the smallest angle.

How this appears as a diagram:

7x 7x

x

The working required:

15x = 180°

x = 12°

x = 12° and 7x = 84°

7x + 7x + x =15x

÷ 15

Isosceles Triangles

Find the value of the pronumeral

3

d 61°

n

71°

8

9

y

53°

2

e

79°

4

k 9°

5

q 45°

6

33°

x

7

f

85°

1 s

48°

s = 180° -

Find the value of the letter, these will take 3 lines to solve.

Example 2m = 180° - 52°

2m = 128°

m = 64° m

52°

m

m+m=2m 10

h

40°

11

c

28°

12

y

122°

13

g

34°

14

j

15

136°

i

16

p

8°

17

An isosceles triangle has one angle of 102°, find the size of the other angles

18

An isosceles triangle has a pair of angles of 17°, find the size of the other angle

19


These are harder!

20 An isosceles triangle has a pair of angles that are twice the size of the other angle find the angles (Hint: use x and 2x)

x = 2x =

21 Repeat the above question only this time the angles are 4 times the size of the other.

Solve these, you might not always need 3 lines. Use 'x=' in your working.


Isosceles Triangles

Find the value of the pronumeral

3

d = 180° - 61° - 61°

d = 58°

d 61°

61°

n

71° 71°

8

n = 180° - 71° - 71°

n = 38°

9

y = 180° - 53° - 53°

y = 74° y

53° 53°

2

e = 180° - 79° - 79°

e = 22° e

79° 79°

4

k = 180° - 9° - 9°

k = 162°

k 9°

9°

5

q = 180° - 45° - 45°

q = 90°

q 45°

45°

6

x = 180° - 33° - 33°

x = 114°

33°

x 33°

7

f = 180° - 85° - 85°

f = 10°

f

85°

85°

1 s

48° 48° s = 84°

s = 180° - 48° - 48°

Find the value of the letter, these will take 3 lines to solve.

Example 2m = 180° - 52°

2m = 128°

m = 64° m

52°

m

m+m=2m 10 2h = 180° - 40°

2h = 140°

h = 70° h

40°

h

11

2c = 180° - 28°

2c = 152°

c = 76° c c

28°

12 2y = 180° - 122°

2y = 58°

y = 29° y

122°

y

13 2g = 180° - 34°

2g = 146°

g = 73° g

34°

g

14 2j = 180° - 90°

2j = 90°

j = 45° j

j

15 2i = 180° - 136°

2i = 44°

i = 22° 136°

i

i

16 2p = 180° - 8°

2p = 172°

p = 86° p

8°

p

17

2x = 180° - 102°

2x = 78°

x = 39°

An isosceles triangle has one angle of 102°, find the size of the other angles

18

x = 180° - 17° - 17°

x = 146°


19

x = 180° - 55° - 55°

x = 70°


These are harder!

20 An isosceles triangle has a pair of angles that are twice the size of the other angle find the angles (Hint: use x and 2x)

5x = 180°

x = 36°

x = 36° 2x = 72°

21 Repeat the above question only this time the angles are 4 times the size of the other.

9x = 180°

x = 20°

4x = 80° x = 20°

Solve these, you might not always need 3 lines. Use 'x=' in your working.


Exterior Angles of a Triangle There is a relationship between the exterior angle of a triangle and the 2 opposite interior angles, the sum of the 2 opposite interior angles = the exterior angle. In other words when you add the 2 opposite interior angles you get the exterior angle. The diagram below shows this graphically. It is important that you realise that the two opposite angles aren’t adjacent to (don't touch) the exterior angle. Column 1 Q.1 - 5 require you to add the two interior angles together to get the exterior angle. The adjacent angle is kept out of these questions. An example is at the top of the column. The next 3 questions show that if you have the adjacent internal angle you just take the supplement to get the external angle (subtract from 180°). So if the adjacent angle is one of the 2 angles, then just take the supplement (subtract it from 180°) of the adjacent angle. This is where the most common mistake is made, if the adjacent angle is given don't add it to the other angle to get your answer as you will be wrong. Column 2 works in reverse, you are given the exterior angle and have to calculate the missing interior angle. Do these by subtracting the given interior angle from the exterior angle, see the example at the top of the column. The third column is challenging as it requires you to use your skills with isosceles and equilateral triangles as well as algebra. There is an example below.

180° -

This angle is ignored

Adjacent angle

130° e

Example : Find the angle Because the triangle is isosceles the other base angle is also e

130° e

e

Solution:

2e = 130° (opposite angle sum = exterior)

e = 65°

Exterior Angles of a Triangle

Find the exterior angle for the following triangles.

Example

x = 116°

x = 57° + 59°

x 59°

57°

1

h 56°

42°

2

n

73°

51°

3

s

49°

4

t

14° 28°

5

v

132°

17°

6

b 55°

7

z

86°

41°

8

c 36°

Find the exterior angle by finding the supple-ment, ignore any extra angles

Example

x = 35°

x = 70° - 35° x

35° 70°

9

g

111° 47°

10

k

144°

12 a

35°

121°

11

e

20° 63°

13

q

158°

14

b 117°

47°

15 d

126°

19°

17

n 71°

28°

16

x

138°

Now the exterior angle is given, use subtraction to find the unknown angle.

Use your knowledge of triangle properties to find the value of the angles

18

u

19

k

31°

21 Hint: Find a then b then c

b 110° a

c

22

60° 3a

12a

20

b 128°

23

3a 2a

a


Exterior Angles of a Triangle

Find the exterior angle for the following triangles.

Example

x = 116°

x = 57° + 59°

x 59°

57°

1

h = 98°

h = 56° + 42°

h 56°

42°

2

n = 124°

n = 51° + 73°

n

73°

51°

3

s = 139°

s = 90° + 49° s

90°

49°

4

t = 42°

t = 14° + 28°

t

14° 28°

5

v = 149°

v = 132° + 17°

v

132°

17°

6

b = 125°

b = 180° - 55°

b 55°

7

z = 139°

z = 180° - 41° z

86°

41°

8

c = 144°

c = 180° - 36°

c 36°

Find the exterior angle by finding the supple-ment, ignore any extra angles

Example

x = 35°

x = 70° - 35° x

35° 70°

9

g = 64°

g = 111° - 47° g

111° 47°

10

k = 54°

k = 144° - 90° 90°

k

144°

12 a

35°

121° a = 86°

a = 121° - 35°

11

e = 43°

e = 63° - 20°

e

20° 63°

13

q = 68°

q = 158° - 90°

q

90° 158°

14

b = 70°

b = 117° - 47°

b 117°

47°

15 d

126°

19° d = 107°

d = 126° - 19°

17

n = 43°

n = 71° - 28°

n 71°

28°

16

x = 48°

x = 138° - 90°

x

138°

90°

Now the exterior angle is given, use subtraction to find the unknown angle.

Use your knowledge of triangle properties to find the value of the angles

18

u = 120°

u = 60° + 60°

u

60°

60° 60°

19

k = 62°

k = 31° + 31° k

31°

31°

21 Hint: Find a then b then c

c = 110° - 70°

a = 70°

a = 180° - 110°

b = 70° (isos)

c = 40°

b 110° a

c

22

60° 3a

12a

3a

3a = 30°

6a = 60°

3a + 3a = 6a

a = 10°

12a = 120°

20

b = 52°

b = 180° - 128°

b 128° b

23

2a = 60°

3a = 90°

3a = 180° - 90°

a = 30°

3a 2a

a

90°


Angles in Quadrilaterals The angle sum of a quadrilateral is 360°, this means that when all the angles inside the shape are added, they total 360°. If you are given 3 angles then to find the missing angle subtract the 3 angles from 360° and what is left is the your answer. In column 1 through to question 11 the exercises are all done the same way, you are given 3 angles (remember a right angle is 90°), subtract these from 360°. If you write in ‘90°’ in the space next to the right angle sign you have less chance of forgetting it in your calculations. The question at the top of the 1st column shows the setting out for the exercises. The rest of column 2 involves a small amount of algebra. The exercises are done in the same way as the first column in that the missing angles equal 360° minus the given angles. Because in the questions the 2 unknown angles are the same then you can add the letters together, i.e. a + a = 2a, x + x = 2x. An example is below. Question 20 is a similar style of question. Column 3 is the same as the first column only one of the angles given is an exterior angle. To change the exterior angle to an interior angle subtract it from 360°. Then you have 3 interior angles which subtracted from 360° gives you the missing angle. The most common mistake is forgetting to change the exterior angle it to an interior angle in these questions.

e

80° e

60°

2e = 360° - 80° - 60°

2e = 220°

e = 110°

Solution: Example: Find the value of e

e + e = 2e

Angles in Quadrilaterals

Find the missing angle in these quadrilaterals

1 x = 360° - 90° - 90° - 90°

x =

x

2

a

3

115° 65°

65° d

8

102°

112°

38°

w

5

43°

v

6

95° 65°

y 105°

7

85°

32° 8° k

4

117° 98°

63° n

9

98°

q 142°

10

m 72°

78°

11

230° 38°

e 32°

These have 2 unknown angles

12

72°

t 118°

t

2t = 360° -

2t =

t =

13

70° m

70° m

14

63°

b

63°

b

15

x

84° x

46°

Find the 2 angles

Convert the exterior angles to interior angles to find x then solve for y.

16

40° 52° x

155°

y x = 360° -

x =

y =

y =

17

x 40° 125°

62°

y

18

x

68° 142°

292°

y

20

3x

120°

2x

3x =

2x =

19

35° x

y 27°


Angles in Quadrilaterals

Find the missing angle in these quadrilaterals

1 x = 360° - 90° - 90° - 90°

x = 90°

x 90°

90° 90°

2 a = 360° - 90° - 90° - 90°

a = 90° a 90°

90° 90°

3

115° 65°

65° d

d = 360° - 65° - 115° - 65°

d = 115°

8

102°

112°

38°

w

w = 360° - 112° - 38° - 102°

w = 108°

5

43° 90°

v

90°

v = 360° - 43° - 90° - 90°

v = 137°

6

95° 65°

y 105°

y = 360° - 105° - 65° - 95°

y = 95°

7

85°

32° 8° k

k = 360° - 8° - 32° - 65°

k = 255°

4 n = 360° - 98° - 117° - 63°

n = 82° 117° 98°

63° n

9 q = 360° - 90° - 142° - 98°

q = 30°

98°

q 142° 90°

10 m = 360° - 78° - 90° - 72°

m = 120° m 72°

78° 90°

11 e = 360° - 38° - 32° - 230°

e = 60° 230° 38°

e 32°

These have 2 unknown angles

12

72°

t 118°

t

2t = 360° - 118° - 72°

2t = 170°

t = 85°

13 2m = 360° - 70° - 70°

2m = 220°

m = 110°

70° m

70° m

14 2b = 360° - 63° - 63°

2b = 234°

b = 117°

63°

b

63°

b

15 2x = 360° - 84° - 46°

2x = 230°

x = 115° x

84° x

46°

Find the 2 angles

Convert the exterior angles to interior angles to find x then solve for y.

16

40° 52° x

155°

y x = 360° - 155°

x = 205°

y = 360° - 205° - 52° - 40°

y = 63°

17 x = 360° - 125°

x = 235°

y = 360° - 235° - 40° - 62°

y = 23°

x 40° 125°

62°

y

18 x = 360° - 292°

x = 68°

y = 360° - 68° - 68° - 142°

y = 82°

x

68° 142°

292°

y

20

90°

3x

120°

2x

5x = 360° - 120° - 90°

5x = 150°

x = 30° 3x = 90°

2x = 60°

19 x = 360° - 90°

x = 270°

y = 360° - 270° - 27° - 35°

y = 28°

35° x

y 27°

90°


ANGLES - Picnic Point High School · 2020-04-02 · Naming Angles Naming angles allows you to explain which angle or part of a shape you're dealing with. The first column introduces

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