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Typesof Triangles

Look at the picture of a sailing boat above. In

the picture, the sails are in the shape of quadrilaterals

and triangles. What is the name of the quadrilateral?

Look at the triangular sails. What types of triangles are they?

1. TYPES OF TRIANGLES BASED ON THE LENGTHS OF

THE SIDES

Look at the Figure 9.1

a. Use a ruler to measure the lengths of the sides of ABC.

What are you going to

9.1

learn? To d etermine the type s

of tr iangle

To find the sum of t he

angles of a tr iangle

To d esc ribe the relat ion

be twe en exter ior and

interior ang les

To find t he fo rmula for

the p er imeter and area

of a tr iang leKey Terms: c ongruent sides

equilateral tr iangle

isosce les tria ng le

scalene tr iangle

r ight tr iang le

obtuse tr iangle

acute tr iangle

exterior ang le

interior a ngle

P

QR

F

EDA B

(a) (b) (c)

Figure 9.1

C

Mathematics for Junior High School Year 7 / 339

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b. Are there any equal sides? If so, how many are they?

c. By looking at the lengths of the sides, what kind of

triangle is ABC? Explain it.

d. Repeat questions (a) to (c) for DEF.

e. Repeat questions (a) to (c) forPQR .

Equila teral triang le is a tr iang le ha ving t hree c ong ruent

sid es.

Isosceles triangle is a triang le ha ving two c ong ruent sides.

Scalene triangle is a t riang le having no c on g ruen t side s.

Relation to the Real World

(a) (c)(b)

Figure 9.2

Look at Figure 9.2 (a), what shape is the high building?

In Figure 9.2 (b), what shape is the sail of the motorboat?

Look at Figure 9.2 (c), what shape is the sail of the

fishermans boat?

Think and Discuss

Look at the figure on the right.

How many equilateral triangles

are there?

/ Students Book Triangles340

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K

L M

Look at isosceles triangle KLMon the left.

Kis called the vertex angle.

L and Mare called the base angles.

2. TYPES OF TRIANGLES BASED ON THE MEASURES OF

THE ANGLES

You have learnt about a rectangle and its properties. To

remind you, discuss the following.

GROUP WORK

1. Draw rec tang le ABCD8 cm in length and 6 cm in width.

2. Draw the d iago nals.3. Cut the recta ngle a long the side a.

4. Cut i t a long one o f the d iag onals.

5. What is the shap e o f the frag me nts?

6. Do bo th frag ments have the sam e a rea?

7. Loo k at b oth frag me nts. Is there one right a ngle in eac h

frag me nt? If so, how do you me asure the ang le? Show

the p osition of the a ngle and me ntion its nam e.

If rectangle ABCD is cut into two parts along one of the

diagonals, we will get two congruent triangles. Because one of

the angles in each triangle is a right angle (C or B), the

triangle is called a right triangle.

L

(c)

A

BC

K R

MP

Q(a) (b)

Figure 9.3

Look at the above figure. Use a protractor to answer the

following questions.

a.Find the measures of the angles of ABC.

b.Is there any 90 angle?

c.What are the measures of the other two angles?


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d.By observing the measures of the angles, what kind of

triangle is ABC? Explain it.

e.Repeat questions (a) to (d) for KLM.

f. Repeat questions (a) to (d) for PQR .

A righ t triang le is a tria ng le tha t ha s on e 90 angle.

An obtuse triangle is a tria ngle tha t ha s one ob tuse a ng le.

An ac ute triangle is a triang le tha t ha s three a c ute

Related to the Real World

(b)(a)

In figure (a), what shape is the sail of the boats?

In figure (b) , what shape is the wooden terrace of the

house?

Look at the following figures.

Arranging Two Isosceles Triangles

Arrange two equal isosceles triangles by attaching the equal

sides.

Draw each geometrical shape you get.

How many kinds of geometrical shape can you get?

What are the names of the shapes?

A B A

C D CD

Figure 9.4

B


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3. TYPES OF TRIANGLES BASED ON THE ANGLES AND SIDES

R

QPC

B

A M

L

K(b) (c)(a)

Figure 9.5

Look at the above figure. Use a protractor and a

ruler to answer the following questions.

a. Find the measures of the angles ofABC.

b. Measure the lengths of the sides of ABC.

c. Are there any equal sides in ABC?

d. Is there any 90 angle in ABC?

e. What are the measures of the other two angles?

f. By observing the angles, what shape is ABC?

g. By observing the lengths of the sides, what shape is

ABC?

h. By observing the angles and the sides, what shape is

ABC? Explain it.

i. Repeat questions (a) to (h) for KLM.

j. Repeat questions (a) to (h) for PQR .

k. Can you find other types of triangle in this grouping?

Explain it.

An obtuse isosceles triangle is a triangle that has oneobtuse angle and two equal sides.

An acute isosceles triangle is a triangle tha t has one acute

angle and two equal sides.

A right isosceles triangle is a triangle that has one 90

angle and two equal sides.


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CRITICAL THINKING

Is it possible for a triangle to have two right angles? Explain.

Is it possible for a triangle to have two obtuse angles? Explain.

Discuss

Music. Have you ever seen a piano like in

the picture on the left? A supporting stick

supports the top of the piano. The stick

forms a 57 angle with the base of the piano,

whereas the top forms a 90 angle with the

stick. How many degrees is the other angle?

57

80

R

Q? ?

2 cm 2 cmPROBLEM 1

Given PQR as in the figure on the right.

a. What kind of triangle is PQR?

Explain it. P

b. What is the measure of P?

c. What is the measure of Q?

d. How can you determine the measures of P and Q?

e. Is P = Q? Why?

PROBLEM 2

Calculate the measure ofeach angle of ABC.

What is the sum of A and C?

Explain.

(4x + 7)

(8x 1)

C B

A


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PROBLEM 3

Look at the figure of FGH on the right.

a. Find the measure of each angle

indicated by x, y, z.b. Based on the measures of the

angles, what kind of triangle is

FGH? F HJ

21

yx65

39

G

z

c. Based on the measures of the

angles, what kind of triangle is

GHJ?

d. Based on the measures of the angles, what kind of triangle

is FGJ?

The Exterior and Interior Angles of a Triangle

An exterior angle of a triangle is an angle formed by the

side of a triangle with the extension of the other side of the

triangle. Think about the definition of the interior angle of a

triangle.

Look at XYZ on the left.

The side XYis lengthened into WY.Y, Z, and YXZ are the interior angles of XYZ

and WXZ is the exterior angleofYXZ .

a. What conclusion can you draw about the relation

between WXZ and YXZ?

b. What is the measure of WXZ?

c.What conclusion can you draw about the relation between the

size of the exterior angle (WXZ) and two interior angles

(XYZ and YZX)?

d.How many exterior angles are there in a triangle?

W

c

YX

ba

Exterior

Angle

Z


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PROBLEM 4

Look at the figure on the right.

a. Name the exterior angle of DEF.

b. What is the measure of the exterior angleof DEF?

35

F

E

D

75G

c. What is the measure of DFE.

d. Measure the measure of EDF.

Based on the above explanation, you can draw the

conclusion as follows:

The me a sure of a n e xterior ang le of a triang le is eq ua l to

the sum of two interior angles which are notsupp lem enta ry to the e xterior ang le.

The Perimeter and Area of a Triangle

To find the perimeter of a triangle, you should know first

the lengths of the sides of the triangle because the perimeter of a

triangle is the sum of the lengths of the sides forming thetriangle.

PROBLEM 5

Look at the figure on the left.C

BA

a.How can you calculate the perimeter of ABC shown

in the figure on the left? Explain.

b.What conclusion can you draw?

c.Can you formulate the perimeter of ABC?


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7 cm

P

R 5 cmQ

If Pis the p erime ter of a triang le h aving the sides a, b d a n c,

then the p erime ter of the triang le c an b e state d as

P = a + b + c

Think and Discuss

Garden Problem

6 m

10 m8 m

14 cm

Pak Budi has a garden as shown in the figure on the

left. He wants to build a fence around the garden.

a. How can you calculate the perimeter of Pak Budis garden?

b. How long is the fence that Pak Budi needs?

c. What is the relation between the perimeter of the garden and

the cost? Explain.

d. If the cost of setting the fence is Rp 25.000,00 per meter, how

much money will Pak Budi spend?

Check Your Understanding

1. Calculate the perimeter of the isosceles triangle

PQR on the right.

2. Explain how you calculate the perimeter of PQR .

WORK IN GROUPSMaterials: grid paper, a ruler, and scissors

1. Draw rectangle ABCDon the grid paper with the length of 12 squares andthe width of 9 squares.

2. Cut the rectangle ABCDalong the sides.3. What is the area of rectangle ABCD?

4. Draw one of the diagonals of rectangle ABCD.5. Cut the rectangle ABCDalong the diagonals (step 4) into two parts.6. What is the shape you get? Are the two shapes obtained equal?7. Do the two shapes have the same area?8. What is the area of each shape you get (step 7)?9. What is the formula of each of the shapes you get?

Mini - Lab


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Critical Thinking

a

t

C

B

x

y

A

Look at ABCin the figure on the right.

a. Calculate the area of ABC.

b. Is there any other way to calculate the

area ofABC?

c. What conclusion can you draw?

The results of the Mini Lab above show that the area of a

triangle can be obtained from the area of a rectangle; the area of

a right triangle is half of that of a rectangle. Therefore, it can be

concluded that:

If A is the a rea of a triang le w hose b a se

is b and height is h, then the a rea of the

triang le c an b e state d a s:

A =2

1(b h)

b

a

h

Think and Discuss

6 m

8 m

Woodwork5 m

A carpenter is going to make a wooden wall

for the back of a warehouse. If the price of

wood is Rp 5,000.00/m2, how much is the

cost to make the wall of the warehouse?

14 c m

13 c m

15 c m

L

K

M

12 c mN

Checking Understanding

Given KLMas shown in the figure on

the right. Calculate the area of KLM.


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1. Name the types of the following triangles. Use a ruler and a

protractor if necessary.

A

B

C

E

F H

D

G I

L

M

K

P

Q

R

O

T

S

V

U

a. What kind of triangle is ABC? Explain it.

b. What kind of triangle is DEF? Explain it.

c. What kind of triangle is GHI? Explain it.

d. What kind of triangle is KLM? Explain it.

e. What kind of triangle is PQR? Explain it.f. What kind of triangle is STO? Explain it.

g. What kind of triangle is UVW? Explain it.

2. Draw the following triangles (if possible) on dotted paper:

a. a triangle with three acute angles.

b. a triangle with one right angle.

c. a triangle with one obtuse angle.

d. a triangle with one right angle and one obtuse angle.

e. a triangle with three different lengths of its sides.

f. a triangle with two equal lengths of its sides.

g. a triangle with three equal lengths of its sides.


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3.Look at the flag of Jamaica onthe right.

a.Name the kinds of triangle in the

picture of the flag of Jamaica.

b.How many triangles are alike on the flag?

Show them.

c.Which triangles have the same

size?

4.Look at the two triangle rulers on the rightfigure.

a. Do they have similarity? Explain it.

b. What are the differences between them?

Explain.

AB

C5.Look at ABC on the right figure.

a.What kinds of triangles form ABC?

b.How many congruent triangles are there in ABC?

c. How many right-angled triangles are there?

d.How many congruent isosceles triangles

are there in ABC?

e.How many isosceles triangles are there in ABC?

f. Is there any equilateral triangle in ABC?


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6. Draw the following triangles. If the triangle cannot be drawn, then

write impossible and state your reasons.

a. a right triangle

b. a right isosceles triangle

c. an obtuse isosceles triangle

d. an obtuse equilateral triangle

7. Critical thinking.

a.Can an isosceles triangle be an acute triangle? Explain.

b.Can a scalene triangle be an acute triangle? Explain.

c.Can a right triangle be an acute triangle? Explain.

d.Can an equilateral triangle be an acute triangle? Explain.

e.Can an acute triangle be an equilateral triangle? Explain.

f. Can a scalene triangle be an obtuse triangle? Explain.

8.Look at the figure on the right.

a.How many congruent triangles are

there?

H D

CBA

O

G F E

b.What kinds of triangles form square

ACEG?

c. How many degrees is each of the

base angles of each triangle?d.How many degrees is the top angle

of each triangle?

e.What angle has the equal size to OAB?

f. How many degrees is OAB?

9.What kind of triangle has the following angles?

a. 90, 40, 50 b. 115, 30, 35 b. 38, 72, 70

10. Given a triangle with two of its angles are 35 and 50. What

is the measure of the third angle? What kind of triangle is it?


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11.Look at the following figures.

a. Measure the unknown angle in each of the triangles above.

K

E

F

D

CB

A ML

10545

I

H45 G

??

?

?

80

50

40

6060

b. Based on the sizes of the angles, what kind of triangle is

each triangle above?

12.Find the sizes of the unknown angles in the isosceles

triangles below.

52

63

47

3 3

2 2 5

5

?

?

?

?

?

?

R

E

BA

Q

P

FDC

13.Look at the figures below.

a. Find the sizes of the unknown angles.

b. What kind of triangle is each triangle on the right?

c.What is the sum of the acute angles of each triangle on the

right?

d.What is the relation between both acute angles in each of

the above triangle?

30

(i)

45

(ii)

35

(iii)


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14.Find out the value of a, b, and c of each of the following

triangles.

3a

352a

(i)

3cc

c

(iii)

2b

(ii)

2b

2b

15.Critical Thinking.

Given a triangle whose angles are 50, 60 and 70.

a.What kind of triangle is it? Why?

b.Can you classify the triangle based on the lengths of thesides? Explain it.

16.Open Question.

Given triangle ABC with one of the angles being 18. What

kind of triangle is it? Explain it.

17.Look at the triangles in the figure below.

a.Determine the sizes of the unknown angles.

b.Sort ascending by the sizes of the angles in each

triangle.c.Sort descending by the lengths of the sides in each

triangle.

d.Make a prediction about the relation between the

results of (b) and (c).

(1) 75

45

M

(3)

O

V

U30

T

S

N

Q

(2)

60P

135

H

28

GF

(4)

110


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18.What are the measures of the angles of a triangle if the

lengths of the sides are as follows:

a.AB = 8, BC= 5, andAC= 7.

b.DE = 15, EF= 18, and DF= 5.

c.XY= 2, YZ = 4, and XZ = 3.

19.What are the lengths of the sides of a triangle if the measures

of the angles are as follows:

a. S = 90, R = 40, T= 50

b. A = 20, B = 120, C= 40

c. X= 70, Y= 30 , Z = 80

d. D = 80, E = 50, F= 50

20.Observation. Is it possible to form a triangle from the

following midribs of palm leaves if the lengths are as the

following? State the reasons.

a.11 cm, 12 cm and 15 cm. c. 6 cm, 10 cm, and 13 cm.

b.2 cm, 3 cm and 6 cm. d. 5 cm, 10 cm and 15

cm.

21.The perimeter of a quadrilateral

PQRS in the figure on the right

is 22 m.

a.Determine the lengths of

PQ, SR, PS and RQ.

b.How can you calculate the

area of PQRS?

c.What is the area of PQRS?

S R

QP

22.Look at the figure below. Which shape has the largest area?

Explain.

(c)(b)

(a)


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23.The perimeter of KLMis 40 cm.

a.What shape is KLM?

b.Determine the lengths of the sides of KLM.

24.Given the shapes below.

xx

III

a. Determine the area of each shape.

b. Which shape has the largest area?

25.Reni has a piece of square patterned

cardboard forming a rectangle with the

length of 25 cm. She is going to make a toy

the shape of which is shown in the figure

on the right. What is the area of the unused

cardboard?

26.Critical Thinking.

Look at the areas oftriangles I and II.Compare the area I withthat of II. Explain.

L

2x 5

xMK

(c)(b)(a)

25 cm


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BUKSIS-9

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