Lesson The Triangle-Sum Vocabulary 14-7

Lesson The Triangle-Sum Property

Chapter 14

14-7

BIG IDEA In all triangles, the sum of the measures of the three angles is the same number.

Ancient Babylonians realized that each night at the same time, stars are in a slightly different position than the night before. They have rotated a small amount. In a year, they rotate all the way to their original position, because Earth goes around the sun in a year. So, they rotate 1 _ 365 of the way around a circle each day. The number 365 is not an easy number to use in making calculations. Because 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 12, it is an easier number to use. In fact, 360 is the smallest number divisible by all these numbers. So the Babylonians thought of the stars as moving 1 _ 360 of the way around the sky each day. From this we get that one full revolution is a rotation of magnitude 360º.

MATERIALS thin paper such as tracing paper, a pencil, and a ruler

Step 1 Draw any triangle ABC and trace it onto the thin paper. Label the angles 1, 2, and 3 at points A, B, and C, as shown.

SMP10_A_SE_C14_L07A_T_002

1

23

A

BC

Step 2 Measure to � nd the midpoint M of BC. Rotate �ABC 180º about M. (Move the triangle you originally drew and trace it again.) The preimage and image are shown. A' is the image of A. When you perform this rotation, where are the images of B and C?


1

1

2 23

3

A

A'

B MC

14-7The Triangle-Sum Property

Chapter 14

Activity

Vocabularyrevolution

interior angles of a polygon

exterior angles of a polygon

36 Some Important Geometry Ideas

Step 3 Measure to � nd the midpoint N of A'C. Rotate �A'BC 180º about N. B' is the image of B.


1

1

12 2

2

3

3

3

A

A'B'

B MN

C

Step 4 Measure to � nd the midpoint O of B'C. Rotate �A'B'C 180º about O. A* is the image of A'.

1

11

12 2

2

2

33

3

3

A

A'

A*

B'

B MN

O

C

Step 5 Measure to � nd the midpoint P of A*C. Rotate �A*B'C 180º about P. B* is the image of B'.


11

11

12 2

2 2

2

33

33

3

A

A'

A*

B*

B'

B MN

O

PC

Step 6 Measure to � nd the midpoint Q of B*C. Rotate �A*B*C 180º about Q. Where is the image of A*?


11

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12 2

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2

2

2

3

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A'

A*

B*

B'

B MN

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P

Q

C

You should � nd that the triangles completely cover the region around point C. Is this result the same for your classmates who started with different triangles?

The Triangle-Sum Property 37

Lesson 14-7

Chapter 14

In the Activity, notice that at point C, ∠1, ∠2, and ∠3 each appear twice. Together, these six angles form a full revolution around point C. One full revolution measures 360º. The Activity illustrates that 2(m∠1 + m∠2 + m∠3) = 360º, so m∠1 + m∠2 + m∠3 = 180º. This is a property of any triangle.

Triangle-Sum Property

The sum of the measures of the three angles of any triangle is 180º.

QY

Example 1 Suppose two angles of a triangle measure 36.4º and 64.6º. What is the measure of the third angle of the triangle?

Solution Let x be the measure of the third angle. The sum of the measures of the three angles is 180º, so

xº + 36.4º + 64.6º = 180º.

x + 101 = 180

x = 79

The third angle measures 79º.

Applying the Triangle-Sum PropertyRecall that complementary angles are two angles whose measures add to 90º. In the diagram of Example 2, ∠TOP and ∠TOS are complementary angles, because m∠POS = 90º.

Example 2In the diagram at the right, � nd x.

Solution x = m∠OTS, and ∠OTS is in �OTS. So x + y + m∠TOS = 180°. m∠TOS = 43°. To fi nd y, use �POS.

m∠P + m∠POS + y = 180°

19° + 90° + y = 180°

y = 71°

So x + 71° + 43° = 180°

x = 66°

QY

What is the sum of the measures of the angles of an isosceles triangle?

QY


QY


SMP08TM2_SE_C06_T_0287

P

T

SO

x

y43º

19º


P

T

SO

x

y43º

19º

Chapter 14


Angles formed by the sides of a polygon are its interior angles. Angles that form linear pairs with interior angles of a polygon are exterior angles of the polygon. In the diagram in Example 3, ∠ ABC, ∠BCA, and ∠CAB are interior angles. ∠DAC, ∠CBE, and ∠ACF are exterior angles.

Example 3 Find the measure of ∠DAC in the drawing at the right.Solution Let x be the measure of ∠DAC. Let y be the

measure of ∠BAC. Use the Triangle-Sum Property to � nd y.

23° + 120° + y° = 180°

143 + y = 180 Arithmetic

y = 37 Add –143 to both sides.

Now use the fact the ∠DAC and ∠BAC form a linear pair.

x° + y° = 180°

x + 37 = 180 Substitution

x = 143 Add –37 to both sides.

So, m∠DAC = 143°.

Notice ∠DAC = m∠ABC + ∠BCA. This is an example of the Exterior Angle Theorem for Triangles.

Exterior Angle Theorem for Triangles

In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle.


23º

120º

AD

CF

B Exº yº


23º

120º

AD

CF

B Exº yº

Lesson 14-7


Chapter 14

Example 4 Use the information given in the drawing. Explain the steps needed to � nd m∠9.


105˚ 21 3

5 74 6

8

9

Solution The drawing is complicated. Examine it carefully. You may wish to make an identical drawing and write in angle measures as they are found. Here is one way to solve the problem.

The arrows indicate that two of the lines are parallel. ∠5 and the angle that measures 105° are corresponding angles. So m∠5 = ? . ∠7 forms a linear pair with ∠5, so they are supplementary. Thus m∠7 = ? . The triangle has a 90° angle created by the perpendicular lines, but it also has ∠7, a ? angle. The Triangle-Sum Property says that the measures of the three angles add to 180°. Therefore, m∠8 = ? . Finally, ∠9 and ∠8 form a linear pair, so m∠9 = ? .

QuestionsCOVERING THE IDEAS

1. At the right is the result of Step 2 in the Activity. Why do the angles of �A'BC have the same measures as the angles of �ABC?

2. Multiple Choice Which is true? A In some but not all triangles, the sum of the measures of the

angles is 180º. B In all triangles, the sum of the measures of the angles is 180º. C The sum of the measures of the angles of a triangle can be any

number from 180º to 360º. D If two angles of a triangle are complementary, then the measure of

the third angle must be greater than 90º.

In 3–6, two angles of a triangle have the given measures. Find the measure of the third angle. 3. 45º, 45º 4. 2º, 3º 5. 70º, 80º 6. xº, 120º - xº

GUIDEDGUIDED


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A'

B MC


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Chapter 14


7. In Example 3, fi nd a. m∠ ACF. b. m∠FBE. c. m∠ ACF + m∠FBE + m∠CAD.

8. In Example 4, explain how to fi nd m∠4.

APPLYING THE MATHEMATICS

9. Copy the fi gure at the right. a. If m∠1 = 40º and m∠2 = 60º, fi ll in the measures of all the

angles in the six triangles. b. Give 5 pairs of parallel lines. c. How do you know the lines are parallel?

10. Many quilt patterns use triangles. In the isosceles triangle outlined in the kaleidoscope quilt block at the right, the vertex angle equals 54º. What is the measure of each base angle?

11. Find m∠DAB below. 12. Find m∠VYZ below.


50º

43º

B

CD A


W

VY Z

50º

13. Three lines intersect forming ∠1, ∠2, ∠3, and ∠4 as shown below. Explain why m∠1 + m∠2 + m∠3 = 180º.


13 4

2

14. Use the information given in the drawing at the right. Find the measures of angles 1 through 8.

15. Explain why a triangle cannot have two right angles.

16. In an equilateral triangle, all the angles have the same measure. What is this measure?


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23

A

C

E

F

D

B

G


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54º54º


123º 61º1

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62 5

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8


123º 61º1

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Lesson 14-7


Chapter 14

17. In the fi gure below, −−−

BA ⊥ −−−

AC . Angle BAC is bisected (split into two equal parts) by

�� AD , and m∠B = 70º. Find the measures of ∠1,

∠2, and ∠3.


BD

CA

12

345º45º

70º

18. Multiple Choice Two angles of a triangle have measures a and b degrees. The third angle must have what measure?

A 180 - a + b degrees B 180 + a + b degrees C 180 + a - b degrees D 180 - a - b degrees

19. In the fi gure at the right, −−−

PQ � −−

RS = T. Find the measures of as many angles in the fi gure as you can.

20. Explain why the measure of an exterior angle must be equal to the sum of the measures of the interior angles at the other two vertices of the triangle.

21. a. Write a theorem about the sum of the measures of the exterior angles of a triangle.

b. Explain why your theorem is true.

EXPLORATION

22. You have seen that the sum of the measures of angles in a triangle on a plane is 180º, but what about a triangle drawn along the surface of a globe or other sphere? Suppose you draw a triangle by starting at the North Pole and drawing a line south. Then draw a line west, then another line north back to the pole. Will the sum of the angles of a triangle on a sphere equal 180º, be greater than 180º, or be less than 180º? Experiment using a globe or other sphere and summarize your results.


P

S

R

Q

T20º 22º

58º 55º


P

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T20º 22º

58º 55º

QY ANSWER

180°


Lesson The Triangle-Sum Vocabulary 14-7

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