2.5 Proving Statements about Segments - Mr Meyers Math

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2.5 Proving Statements about Segments

What you should learn

GOAL 1 Justify statements about congruent segments.

GOAL 2 Write reasons for steps in a proof.

Why you should learn it

Properties of congruence

allow you to justify segment

relationships in real life, such

as the segments in the trestle

bridge shown and in

Exs. 3?5.

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GOAL 1 PROPERTIES OF CONGRUENT SEGMENTS

A true statement that follows as a result of other true statements is called a theorem. All theorems must be proved. You can prove a theorem using a two-column proof. A two-column proof has numbered statements and reasons that show the logical order of an argument.

THEOREM

THEOREM 2.1 Properties of Segment Congruence

Segment congruence is reflexive, symmetric, and transitive. Here are some examples:

REFLEXIVE SYMMETRIC TRANSITIVE

For any segment AB, A?B ? A?B . If A?B ? C? D , then C? D ? A?B . If A?B ? C? D , and C? D ? ? EF , then A?B ? ? EF .

E X A M P L E 1 Symmetric Property of Segment Congruence

STUDENT HELP

Study Tip When writing a reason for a step in a proof, you must use one of the following: given information, a definition, a property, a postulate, or a previously proven theorem.

You can prove the Symmetric Property X

P

of Segment Congruence as follows.

GIVEN P?Q ? XY

PROVE X?Y P?Q

Y

q

Statements 1. P?Q ? XY

2. PQ = XY

3. XY = PQ 4. ? XY P?Q

Reasons

1. Given 2. Definition of congruent segments 3. Symmetric property of equality 4. Definition of congruent segments

You are asked to complete proofs for the Reflexive and Transitive Properties of Segment Congruence in Exercises 6 and 7. . . . . . . . . . .

A proof can be written in paragraph form, called paragraph proof. Here is a paragraph proof for the Symmetric Property of Segment Congruence. Paragraph Proof You are given that P?Q ? ? XY. By the definition of congruent segments, PQ = XY. By the symmetric property of equality, XY = PQ. Therefore, by the definition of congruent segments, it follows that ? XY ? P?Q.

102

Chapter 2 Reasoning and Proof THEOREM 2.1 PROPERTIES OF SEGMENT CONGRUENCE

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GOAL 2 USING CONGRUENCE OF SEGMENTS

E X A M P L E 2 Using Congruence

Proof Use the diagram and the given information to complete

K

the missing steps and reasons in the proof.

J

GIVEN LK = 5, JK = 5, J?K ? JL

PROVE ? LK ? JL

L

Statements

1. a. 2. b. 3. LK = JK 4. ? LK J?K 5. ? JK J?L

6. d.

Reasons

1. Given 2. Given 3. Transitive property of equality 4. c. 5. Given 6. Transitive Property of Congruence

SOLUTION a. LK = 5 b. JK = 5 c. Definition of congruent segments d. ? LK ? ? JL

Proof

STUDENT HELP

Study Tip The distributive property can be used to simplify a sum, as in Step 5 of the proof. You can think of PQ + PQ as follows: 1(PQ) + 1(PQ) = (1 + 1) (PQ) = 2 ? PQ.

E X A M P L E 3 Using Segment Relationships

In the diagram, Q is the midpoint of ? PR.

P

R

Show that PQ and QR are each equal to 12PR.

q

SOLUTION Decide what you know and what you need to prove. Then write the proof. GIVEN Q is the midpoint of ? PR.

PROVE PQ = 12PR and QR = 12PR.

? Statements 1. Q is the midpoint of ? PR. 2. PQ = QR 3. PQ + QR = PR 4. PQ + PQ = PR 5. 2 ? PQ = PR 6. PQ = 12PR 7. QR = 12PR

Reasons 1. Given 2. Definition of midpoint 3. Segment Addition Postulate 4. Substitution property of equality 5. Distributive property

6. Division property of equality

7. Substitution property of equality

2.5 Proving Statements about Segments

103

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ACTIVITY

Construction Copy a Segment

Use the following steps to construct a segment that is congruent to ? AB.

A

B

A

B

C

1 Use a straightedge to draw a segment longer than ? AB. Label the point C on the new segment.

C

2 Set your compass at the length of ? AB.

A

B

C

D

3 Place the compass point at C and mark a second point, D, on the new segment. C?D is congruent to ? AB.

You will practice copying a segment in Exercises 12?15. It is an important construction because copying a segment is used in many constructions throughout this course.

GUIDED PRACTICE

Vocabulary Check

1. An example of the Symmetric Property of Segment Congruence is "If ? AB ? ?, then C?D ? ?."

Concept Check

2. ERROR ANALYSIS In the diagram below, C?B ? ? SR and C?B ? Q?R. Explain what is wrong with Michael's argument.

Because C?B ? S? R and C?B ? Q? R , A then C?B ? A? C by the Transitive

Property of Segment Congruence.

CB

Q SR

Skill Check

BRIDGES The diagram below shows a portion of a trestle bridge, where ? BF fi C? D and D is the midpoint of ? BF .

3. Give a reason why B?D and F?D are congruent.

AC E

4. Are TMCDE and TMFDE complementary? Explain.

5. If C?E and B?D are congruent, explain why C?E and F?D are congruent.

BD F

104 Chapter 2 Reasoning and Proof

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PRACTICE AND APPLICATIONS

STUDENT HELP

Extra Practice to help you master skills is on p. 806.

PROVING THEOREM 2.1 Copy and complete the proof for two of the cases of the Properties of Segment Congruence Theorem.

6. Reflexive Property of Segment Congruence

GIVEN EF is a line segment

PROVE ? EF ? ? EF

E F

Statements

1. EF = EF 2. ?

Reasons

1. ? 2. Definition of congruent segments

7. Transitive Property of Segment Congruence

GIVEN A?B ? J? K , J? K ? S?T

B

PROVE A?B ? S?T

S A

Statements 1. ? AB ? J? K , J? K ? S?T

2. AB = JK, JK = ST

3. AB = ST 4. ? AB ?S?T

Reasons

1. ? 2. ? 3. ? 4. ?

J

T

K

xy USING ALGEBRA Solve for the variable using the given information. Explain your steps.

8. GIVEN ? AB ? B?C, C?D ? B?C

9. GIVEN PR = 46

A 2x 1 B

C 4x 11 D

P 2x 5 q

6x 15

R

10. GIVEN S?T ? ? SR, Q?R ? S?R

S

T

5(3x 2)

11. GIVEN ? XY ? ? WX, ? YZ ? ? WX

X

Y

4x 3

9x 12

q x4 R

W

Z

CONSTRUCTION In Exercises 12?15, use the segments, along with a straightedge and compass, to construct a segment with the given length.

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 6, 7 Example 2: Exs. 16?18 Example 3: Exs. 16?18

A

x

B

C

12. x + y

E

z

13. y ? z

F

14. 3x ? z

y

D

15. z + y ? 2x

2.5 Proving Statements about Segments 105

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FOCUS ON CAREERS

16. DEVELOPING PROOF Write a complete proof by rearranging the

reasons listed on the pieces of paper.

GIVEN U?V ? ? XY, V? W ? W?X, W?X ?? YZ PROVE U? W ? ? XZ

UV

W

XY

Z

Statements 1. U?V ? ? XY, V? W ? W?X,

W?X ?? YZ 2. V? W ?? YZ

3. UV = XY, VW = YZ

4. UV + VW = XY + YZ

5. UV + VW = UW, XY + YZ = XZ

6. UW = XZ 7. U? W ? ? XZ

Reasons Transitive Property of Segment Congruence Addition property of equality Definition of congruent segments Given Segment Addition Postulate

Definition of congruent segments Substitution property of equality

TWO-COLUMN PROOF Write a two-column proof.

17. GIVEN XY = 8, XZ = 8, ? XY ? ? ZY 18. GIVEN N?K ? ? NL, NK = 13

PROVE ? XZ ? ? ZY

PROVE NL = 13

Y

J

K

X

Z

N

M

L

19. CARPENTRY You need to cut ten wood planks that are the same size. You measure and cut the first plank. You cut the second piece, using the first plank as a guide, as in the diagram below. The first plank is put aside and the second plank is used to cut a third plank. You follow this pattern for the rest of the planks. Is the last plank the same length as the first plank? Explain.

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For many projects, carpenters need boards that are all the same length. For instance, equally-sized boards in the house frame above insure stability.

ERNET

CAREER LINK



20. OPTICAL ILLUSION To create the illusion, a U

special grid was used. In the grid, corresponding V

row heights are the same measure. For instance, U?V and ? ZY are congruent. You decide to make this W

design yourself. You draw the grid, but you need to

make sure that the row heights are the same. You measure U?V , U? W, ? ZY , and ? ZX. You find that

X

U?V ? ? ZY and U? W ? ? ZX. Write an argument that allows you to conclude that ? VW ? Y?X .

Y Z

106 Chapter 2 Reasoning and Proof

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