2.6 Properties of Equality and Congruence

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2.6 Properties of Equality

and Congruence

Goal

Use properties of equality and congruence.

Key Words

? Reflexive Property ? Symmetric Property ? Transitive Property

Reflexive Property

Symmetric Property

Jean is the same height as Jean.

If

Jean is the same height as Pedro,

then

Pedro is the same height as Jean.

Transitive Property

If

Jean is the same height as Pedro

and

Pedro is the same height as Chris,

then

Jean is the same height as Chris.

Student Help

LOOK BACK To review the difference between equality and congruence, see p. 30.

The photos above illustrate the Reflexive, Symmetric, and Transitive Properties of Equality. You can use these properties in geometry with statements about equality and congruence.

PROPERTIES OF EQUALITY AND CONGRUENCE

Reflexive Property Equality AB AB

maA maA

Congruence A&B* c A&B*

aA c aA

Symmetric Property

Equality

If AB CD, then CD AB.

Congruence

If A&B* c C&D*, then C&D* c A&B*.

If maA maB, then maB maA. If aA c aB, then aB c aA.

Transitive Property

Equality

If AB CD and CD EF, then AB EF.

If maA maB and maB maC, then maA maC.

Congruence

If A&B* c C&D* and C&D* c E&F*, then A&B* c E&F*.

If aA c aB and aB c aC, then aA c aC.

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EXAMPLE 1 Name Properties of Equality and Congruence

Name the property that the statement illustrates. a. If G&H* c J&K*, then J&K* c G&H*. b. DE DE c. If aP c aQ and aQ c aR, then aP c aR.

Solution a. Symmetric Property of Congruence b. Reflexive Property of Equality c. Transitive Property of Congruence

Name Properties of Equality and Congruence

Name the property that the statement illustrates. 1. If DF FG and FG GH, then DF GH. 2. aP c aP 3. If maS maT, then maT maS.

Logical Reasoning In geometry, you are often asked to explain why statements are true. Reasons can include definitions, theorems, postulates, or properties.

EXAMPLE 2 Use Properties of Equality

In the diagram, N is the midpoint of

M&P**, and P is the midpoint of N&Q*.

M

N

P

P

Show that MN PQ.

Solution MN NP NP PQ MN PQ

Definition of midpoint Definition of midpoint Transitive Property of Equality

Use Properties of Equality and Congruence

4. a1 and a2 are vertical angles, and a2 c a3. Show that a1 c a3.

a1 c a2 a2 c a3

__?__ Theorem Given

1

2

3

a1 c a3

__?__ Property of Congruence

2.6 Properties of Equality and Congruence

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Student Help

STUDY TIP In geometry, you can use properties of equality that you learned in algebra.

IStudent Help



MORE EXAMPLES More examples at

PROPERTIES OF EQUALITY

Addition Property

Adding the same number to each side of an equation produces an equivalent equation.

Example

x37 x3373

Subtraction Property

Subtracting the same number from each side of an equation produces an equivalent equation.

Example

y 5 11 y 5 5 11 5

Multiplication Property

Example

Multiplying each side of an equation by the same nonzero number produces an equivalent equation.

14z 6 14z p 4 6 p 4

Division Property

Example

Dividing each side of an equation by the same nonzero number produces an equivalent equation.

8x 16 88x 186

Substitution Property

Substituting a number for a variable in an equation produces an equivalent equation.

Example

x7 2x 4 2(7) 4

EXAMPLE 3 Justify the Congruent Supplements Theorem

a1 and a2 are both supplementary to a3. Show that a1 c a2.

1

2

3

Solution ma1 ma3 180 ma2 ma3 180 ma1 ma3 ma2 ma3 ma1 ma2 a1 c a2

Definition of supplementary angles Definition of supplementary angles Substitution Property of Equality Subtraction Property of Equality Definition of congruent angles

Use Properties of Equality and Congruence

5. In the diagram, M is the midpoint of A&B*. Show that AB 2 p AM.

MB AM AB AM MB

Definition of __?__ __?__ Postulate

A

M

B

AB AM AM

__?__ Property of Equality

AB 2 p AM

Distributive property

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2.6 Exercises

Guided Practice

Vocabulary Check Skill Check

Match the statement with the property it illustrates.

1. maDEF maDEF

A. Symmetric Property of Equality

2. If P&Q* c S&T*, then S&T* c P&Q*. B. Reflexive Property of Equality

3. X&Y* c X&Y*

C. Transitive Property of Equality

4. If aJ c aK and aK c aL, then aJ c aL.

D. Reflexive Property of Congruence

5. If PQ QR and QR RS, then PQ RS.

E. Symmetric Property of Congruence

6. If maX maY, then maY maX.

F. Transitive Property of Congruence

Name the property that the statement illustrates. 7. aABC c aABC 8. If maB maD and maD maF, then maB maF. 9. If G&H** c J&K*, then J&K* c G&H**.

Practice and Applications

Extra Practice

See p. 676.

Homework Help

Example 1: Exs. 10?18 Example 2: Exs. 19?24 Example 3: Exs. 19?24

Completing Statements Use the property to complete the statement. 10. Reflexive Property of Equality: JK __?__

11. Symmetric Property of Equality: If maP maQ, then __?__ __?__.

12. Transitive Property of Equality: If AB BC and BC CD, then __?__ __?__.

13. Reflexive Property of Congruence: __?__ c aGHJ

14. Symmetric Property of Congruence: If __?__ c __?__, then aXYZ c aABC.

15. Transitive Property of Congruence: If G&H** c I*J* and __?__ c __?__, then G&H** c P&Q*.

Naming Properties Name the property that the statement illustrates. 16. If AB CD, then AB EF CD EF.

17. If maC 90, then 2(maC) 15 2(90) 15.

18. If XY YZ, then 3 p XY 3 p YZ.

2.6 Properties of Equality and Congruence

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19. Using Properties In the diagram, ma1 ma2 132, and ma2 105. Complete the argument to show that ma1 27.

ma1 ma2 132 ma2 105 ma1 105 132 ma1 27

Given Given __?__ Property of Equality __?__ Property of Equality

21

20. Using Properties of Congruence In the diagram, A&B* c F&G*, and B^&F( bisects A&C* and D&G**. Complete the argument to show that

B&C* c D&F*.

B&C* c A&B* A&B* c F&G* F&G* c D&F*

Definition of __?__ __?__ Definition of __?__

A

B

D

F

C

B&C* c D&F*

__?__ Property of Congruence

G

21. Unscramble the Steps In the diagram below, PQ RS. Copy the diagram and arrange the statements and reasons in order to make a logical argument to show that PR QS.

P

PR

S

PR = QS

Given

PQ + QR = RS + QR

PQ + QR = PR

Addition Property of Equality

Segment Addition Postulate

RS + QR = QS

PQ = RS

Substitution Property of Equality

Segment Addition Postulate

22. Using Properties of Equality In the diagram at the right, maWPY maXPZ. Complete the argument to show that maWPX maYPZ.

maWPY maXPZ maWPX maWPY maYPX maYPZ maYPX maXPZ maWPY maYPX maYPX maXPZ maWPX maYPZ

WY

X

P

Z

Given __?__ __?__ __?__ __?__

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Student Help

LOOK BACK To review the Congruent Complements Theorem, see p. 69.

23. Congruent Complements Theorem Show that the Congruent Complements Theorem is true. Use Example 3 on page 90 as a model. Provide a reason for each step.

In the diagram, 1 is complementary to 2, and 3 is complementary to 2. Show that 1 c 3.

12

3

24. Error Analysis In the diagram, S&R* c C&B* and A&C* c Q&R*. Explain what is wrong with the student's argument.

Because S&R* c C&B* and A&C* c Q&R*, A

then C&B* c A&C* by the Transitive

Property of Congruence.

CB

Q SR

Using Algebra Find the value of the variable using the given information. Provide a reason for each step.

25. AB BC, BC CD

26. QR RS, ST RS

A 3t 1 B

C 7D

P 23 R

S 5n 2 T

27. Challenge Fold two corners of a piece

of paper so their edges match as shown

1

at the right.

1

What do you notice about the angle

22

formed by the fold lines?

Show that the angle measure is always the same. Provide a reason for each step.

Standardized Test Practice

28. Multiple Choice Which statement illustrates the Symmetric Property of Congruence?

A If A&D* c B&C*, then D&A* c C&B*. B If W&X** c X&Y* and X&Y* c Y&Z*, then W&X** c Y&Z*. C If A&B* c G&H**, then G&H** c A&B*. D A&B* c B&A*

29. Multiple Choice In the figure below, Q&T* c T&S* and R&S* c T&S*.

What is the value of x?

P

R

F 4 H 16

G 12 J 32

7x 4

6x 8

T

S

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Mixed Review

Naming Collinear Points Use the diagram to name a point that is

collinear with the given points. (Lesson 1.3)

B

30. G and E

31. F and B

D F

32. A and D

33. B and D

C

E

G

A

Sketching Intersections Sketch the figure described. (Lesson 1.4) 34. Three lines that do not intersect but lie in the same plane.

35. Two lines that intersect at one point, and another line that intersects both of those lines at different points.

Algebra Skills

Plotting Points Plot the point in a coordinate plane. Then determine which quadrant, if any, the point lies in. (Skills Review, p. 664)

36. (5, 2)

37. (0, 7)

38. (1, 4)

39. (8, 3)

40. (6, 7)

41. (10, 2)

42. (1, 1)

43. (9, 4)

Quiz 2

Find the measures of the numbered angles. (Lesson 2.4)

1. 1

54 3 2

2.

4 140 5

6

3.

10 7 9 8 41

In Exercises 4 and 5, rewrite the statement as an if-then statement. (Lesson 2.5)

4. A square is a four-sided figure. 5. The value of x 2 is 25 if x 5.

6. Use the Law of Syllogism to write the statement that follows from the pair of true statements. (Lesson 2.5)

If we charter a boat, then we will go deep sea fishing.

If we go deep sea fishing, then we will be gone all day.

7. In the diagram, K&M*( bisects aJKN, and K&N*( bisects aMKL. Complete the argument to show that maJKM maNKL. (Lesson 2.6)

maJKM maMKN maMKN maNKL

Definition of __?__ Definition of __?__

M

J

N

maJKM maNKL

__?__ Property of Equality

K

L

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