26 Proving Anglesr Congruent

Mathematics Florida Standards

MAFS.912.G-C0.3.9 Prove theorems about lines and

angles; use theorems about lines and angles to solveproblems. Theorems include;... vertical angles arecongruent...

MP 1, MP 3. MP~4, MP 6

Objective To prove and apply theorems about angles

W I Getting Reodyi X C

A quitter wonts to duplicate this quilt but knows the measure ofonly two ongles. What are the measures of angles 1, 2, 3. and 4?How do you know?

Use what you'velearned about

congruent angle\j)airs.

MATHEMATICAL

PRACTICES

Lesson

Vocabulary• theorem

paragraph proof

In the Solve It, you may have noticed a relationship between vertical angles. You canprove that this relationship is always true using deductive reasoning. A theorem is aconjecture or statement that you prove true.

Essential Understanding You can use given information, definitions,properties, postulates, and previously proven theorems as reasons in a proof.

Theorem 2-1 Vertical Angles Theorem

Vertical angles are congruent.

= 1.3 and Z.2 = ̂4

When you are writing a geometric proof, it may help to separate the theorem you wantto prove into a h>TD0thesis and conclusion. Another way to write the Vertical AnglesTheorem is "If two angles are vertical, then they are congruent." The hypothesis

becomes the given statement, and the conclusion becomes what you want to prove.A two-column proof of tlie Vertical Angles Theorem follows.

120 Chapter 2 Reasoning and Proof

Pi^f Proof of Theorem 2-1: Vertical Angles Theorem

Given: Z.1 and Z.3 are vertical angles.

Prove: = z.3

Statements Reasons

1) Z.1 and Z.3 are vertical angles. 1) Given

2) Z.1 and Z.2 are supplementary. 2) Z that form a linear pair are supplementary.

Z.2 and Z.3 are supplementary.

3) m^l + mZ2=i80 3) The sum of the measures of supplementarymA2 + mA3 = 180 Z is 180.

4) mZ.1 + mZ.2 = mZ.2 + mZ.3 4) Transitive Property of Equality

5) mZ.1 = mZ3 5) Subtraction Property of Equality

6) Z1 = Z3 6) Zi with the same measure are =.

How do you getstarted?

Look for a relationshipin the diagram thatallows you to writean equation with thevariable.

Using the Vertical Angles Theorem

What is the value of x?

The two labeled anglesare vertical angles, so setthem equal.

Solve for X by subtracting2x from each side and

then dividing by 2.

Grid the answer as 21/2

or 10.5.

2x + 21 = 4x

21 = 2x

^ = x2 ̂

2 1 /2

D (OUOUOKO

IGRIDbED RESPONSE

2x + 21)

^ Got It? 1. Wliat is tlie value of x?

CiowerGeometry.com Lesson 2-6 Proving Angles Congruent 121

Proof

Why does theTransitive Propertywork for statements

3 and 5?

In each case, an angle iscongruent to two otherangles, so the two anglesare congruent to eachother.

Problem 2 Proof Using the Vertical Angles Theorem

Given:

Prove:

/.I = /.4

/.2 = ̂3

Statements Reasons

1) zll = Z4 1) Given

2) Z.4 = /-2 2) Vertical angles are =.

3) L.\ = L2 3) Transitive Property of Congruence

4) L\ = A3 4) Vertical angles are =.

5) A2 = Z.3 5) Transitive Property of Congruence

Got It? 2. a. Use the Vertical Angles Theorem to prove the following.Given: = L2

Prove: Z1 = Z2 = ̂3 = Z4

b. Reasoning How can you prove Z.1 = Z2 = zi3 = Z.4without using the Vertical Angles Theorem? Explain.

The proof in Problem 2 is two-column, but there are many ways to display a proof. Aparagraph proof is written as sentences in a paragraph. Below is the proof fromProblem 2 in paragraph form. Each statement in the Problem 2 proof is red in the

paragraph proof.

ProofGiven: l.\ = L4

Prove: Z2 = Z3

Proof: zll = ./4 is given. Z.4 = L2 because vertical angles are congruent.By the Transitive Property of Congruence, Z. I = L2. L.\ = Z.3 becausevertical angles are congruent. By the Transitive Property of Congruence,1.2 = ̂3.

The Vertical Angles Theorem is a special case of the following theorem.

Theorem 2-2 Congruent Supplements Theorem

Theorem

If two angles are supplements

of the same angle (or ofcongruent angles), then thetwo angles are congruent.

If...

Z.1 and Z.3 are supplements and

Z.2 and Z.3 are supplements

Then...

L\ = Z.2

You will prove Theorem 2-2 in f^blem 3.

J

122 Chapter 2 Reasoning and Proof

Proof

How can you use thegiven Information?Both /it and Ll are

supplementary to Z.3.Use their relationshipwith Z.3 to relate Z.1

and Ll to each other.

Writing a Paragraph Proof

Given: L \ and Z.3 are supplementary.Z.2 and Z.3 are supplementary.

-<

Prove: L\ = L2

Proof: z.1 and Z.3 are supplementary because it is given. So mZ.1 + mZ.3 = 180by the definition of supplementary angles. Z.2 and Z.3 are supplementary

because it is given, so mZ.2 + mZ.3 = 180 by the same definition. By the

Transitive Property of Equality, mZ.1 + mZ.3 = mZ.2 + mZ.3. Subtract mZ.3

from each side. By the SubtractionProperty of Equality, mZ.1 = mZ2. Angles

with the same measure are congruent, so Z1 = Z2.

Got It? 3. Write a paragraph proof for the Vertical Angles Theorem.

The following theorems are similar to the Congruent Supplements Theorem.

Theorem 2-3 Congruent Complements Theorem

Theorem If... Then ...

If two angles are L\ and L2 are complements L\ = Z3

complements of the same and Z3 and L.2 are

angle (or of congruent complementsangles), then the two

angles are congruent. \ 3

Theorem 2-4

Theorem

All right angles are

congruent.

If...

L \ and L.2 are right angles

Q—^1

IL

Theorem 2-5 ...

Theorem If...

If two angles are congruent Z1 s Z2, and Z1 and Z2

and supplementary, then are supplements

each is a right angle.

You will prove Theorem 2-3 in Exercise 13.

Then ...

Z1 = Z2

Ybu will prove Theorem 2-4 in Exercise 18.

Then...

mZl = mZ2 = 90

:v

You will prove Theorem 2-5 in Exercise 23.

C PowerGeometry.com Lesson 2-6 Proving Angles Congruent 123

Lesson Check

Do you know HOW?

1. What are the measures of Z.1, L2, and Z.3?

2. What is the value of^:?

C£) 12 CO 120

CD 20 CD 136

(6x + 16)'

MATHEMATICAL

Do you UNDERSTAND? PRACTICES

3. Reasoning If and LB are supplements, and LA

and LC are supplements, what can you conclude

about LB and LCI Explain.

4. Error Analysis Your friend knows that Z.1 andZ.2

are complementary and that Z.1 and Z.3 are

complementary. He concludes that Z.2 and Z.3 must

be complementary. What is his error in reasoning?

5. Compare and Contrast How is a theorem different

from a postulate?

Practice and Problem-Solving Exercises

Practice

6.

^ (80 - xY

Find the value of each variable.

6.

MATHEMATICAL

PRACTICES

^ See Problem 1.

{x + SOYA^^

Find the measures of the labeled angles in each exercise.

9. Exercise 6 10. Exercise 7 11. Exercise 8

12. Developing Proof Complete the following proof by filling in the blanks.

Given: LI = L3

Prove: Z.8 = L4

Statements Reasons

1) Ll = L3 1) Given

2) L3 = L6 2) a. J_

3) b,J_ 3) Transitive Property of Congruence

4) Ll = L4 4) c._^

5) L3 = L4 5)

^ See Problem 2.

I

J

124 Chapter 2 Reasoning and Proof

13. Developing Proof Fill in the blanks to complete this proof ofthe Congruent Complements Theorem (Theorem 2-3).

If two angles are complements of the same angle, then

the two angles are congruent.

Given: L\ and Z.2 are complementary.

Z.3 and Z.2 are complementary.

Prove: ^ Z3

Proof: Z.1 and Z.2 are complementary and Z.3 and Z.2 are complementarybecause it is given. By the definition of complementary angles,

mZ.1 -H mZ.2 = a. _^and mZ.3 -h mZ.2 = b. ? .ThenmZ.1 + mZ.2 = mZ.3 -F mZ.2 by the Transitive Property of Equality.

Subtract mZ.2 from each side. By the Subtraction Property of Equality, youget mZ. 1 = c. Angles with the same measure are d. so Z. 1 = Z.3.

Apply 14. Think About a Plan What is the measure of the angleformed by Park St. and 116th St.?

• Can you make a connection between the angle you

need to find and the labeled angle?

• How are angles tliat form a right angle related?

15. Open-Ended Give an example ofvertical angles in yourhome or classroom.

Park St.

3

4^ See Problem 3.

IL

5'

Algebra Find the value of each variable and the measure of each labeled angle.

x+ 10) (4x - 35)=(3x + 8)° X (5x - 20)°

18. Developing Proof Fill in the blanks to complete this proof of Theorem 2-4.

All right angles are congruent.

Given: Z.X and Z. Y are right angles.

Prove: IX= LY

Proof: Z.X and a. ? are right angles because it is given.By the definition of b. m/.X= 90 and m/LY= 90.

By the Transitive Property of Equality, m/LX = c. _?_.Because angles of equal measure are congruent, d.

19. Miniature Golf In the game of miniaturegolf, the bail bounces off the wall at the

same angle it hit the wall. (This is the angleformed by the path of the ball and the line

perpendicular to the wall at the point ofcontact.) In the diagram, the ball hits thewall at a 40° angle. Using Theorem 2-3,

what are the values of x and y?

X

ovtferGeometrv.com Lesson 2-6 Proving Angles Congruent 125

Name two pairs of congruent angles in each figure. Justify your answers.

23. Developing Proof Fill in the blanks to complete this proof of Theorem 2-5.

If two angles are congruent and supplementary, then each is a right angle.

Given: L.VJ and L.V are congruent and supplementary.

Prove: Z. W and Z. V are right angles.

Proof: Z.W and Z.y are congruent because a. Because

congruent angles have the same measure, mAW =

b. _?_. Z. W and Z. V are supplementary because itis given. By the definition of supplementary angles,

mZ. W + mZ. = c. J_. Substituting mZ. W for mL V,you get mZ. W + mZ. W = 180, or 2mZ. W = ISO. By

the d. ? Property of Equality, mZ.lV= 90. Since

mZ. W = mZ. V, mZ. V = 90 by the Transitive Property

of Equality. Both angles are e. J_ angles by thedefinition of right angles.

24. Design In the photograph, the legs of the table are constructed

sothatz.1 s Z.2. What theorem can you use to justify the

statement that Z.3 = Z.4?

25. Reasoning Explain why this statement is true: Ifm^ABC + m^XYZ = 180 and j^ABC = AXYZ, then AABC

and Z-XVZ are right angles.

W V

Algebra Find the measure of each angle.

26. /LA is twice as large as its complement, Z.B.

28. /LA is twice as large as its supplement, /LB.

27. LA is half as large as its complement, LB.

29. LA is half as large as twice its supplement, LB.

30. Write a proof for this form of Theorem 2-2.

If two angles are supplements of congruent angles,

then the two angles are congruent.

Given: &1 and L2 are supplementary.Z.3 and Z.4 are supplementary.

L2 = LA

Prove: ̂ 1 = ̂3

Challenge 31. Coordinate Geometry LDOE contains points D(2,3), 0(0, 0), and E(5,1)the coordinates of a point F so that OF is a side of an angle that is adjacentsupplementary to LDOE.

. Find

and

126 Chapter 2 Reasoning and Proof

32. Coordinate Geometry Z.AOX contains points/l(l, 3), 0(0,0), andX(4,0).a. Find the coordinates of a point B so that ABOA and /LAOX are adjacent

complementary angles.

b. Find the coordinates of a point C so that OC is a side of a different angle that isadjacent and complementary to /LAOX.

Algebra Find the value of each variable and the measure of each angle.

33. ^ 34. X , ^ 35.

(x + y + 10)°

rStandardized Test Prep SIRIIDbEb RESPONSE

SAT/Aa

r

36. Z.1 and Z.2 are vertical angles. If mAl = 63 and mi:i2 = 4x — 9,

what is the value of x?

37. What is the area in square centimeters of a triangle with a base of 5 cmand a height of 8 cm?

38. In the figure at the right, m/.l = |(mzl2), m/.2 = |(mZ3). IfmZ.3 - 72, what is mZ.4?

39. What is the measure of an angle with a supplement that is four

times its complement?

Mixed Review

Which property of equality or congruence justifies going from the firststatement to the second?

^ See Lesson 2-5.

40. 3x + 7 = 19

3x= 12

41. 4x = 20

x= 5

42. /LI = LI and Z3 s L2

LI = L3

Get Ready I To prepare for Lesson 3-1, do Exercises 43-48.

Refer to the figure at the right.

43. Name four points on line f.

44. Are points G, A, and B collinear?

45. Are points F, 1, and H collinear?

46. Name the line on which point E lies.

47. Name line t in three other ways.

48. Name the point at which lines t and r intersect.

^ See Lesson 1-2.

Lesson 2-6 Proving Angles Congruent 127