4.5 Using Congruent Triangles

Goal 1: Planning a Proof

Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures.

Planning a Proof

For example, suppose you want to prove that  PQS ≅  RQS in the diagram shown at the right.

One way to do this is to show that ∆PQS ≅ ∆RQS by the SSS Congruence Postulate.

use  PQS ≅ the corresponding congruent  Then you can RQS.

fact parts triangles that of are congruent to conclude that P Q S R

Example 1: Planning & Writing a Proof

Given: AB ║ CD, BC ║ AD Prove: AB ≅CD Plan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent.

A B D C

Example 1: Planning & Writing a Proof

Solution: First copy the diagram and mark it with the given information.

Then mark any additional information you can deduce.

Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A B D C

Example 1: Paragraph Proof

Because AD ║CD, it follows from the Alternate Interior Angles Theorem that ≅  CDB.

reason,   ABD For the same ADB because BC ║DA.

≅  CBD By the Reflexive property Congruence, BD of ≅ BD. You can use the ASA Congruence Postulate to conclude that ∆ABD ≅ ∆CDB.

Finally because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD.

A B D C

Example 2: Planning & Writing a Proof

Given: A is the midpoint of MT, A is the midpoint of SR.

Prove: MS ║TR.

Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that  M ≅  T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR.

S M A R T

M R

Given: A is the midpoint of MT, A is the midpoint of SR.

Prove: MS ║TR.

Statements: 1.

A is the midpoint of MT, A is the midpoint of SR.

2. MA ≅ TA, SA ≅ RA 3.

4.

 MAS ≅ 5.

 M ≅  T 6. MS ║ TR  TAR ∆MAS ≅ ∆TAR Reasons: 1.

Given S A 2. Definition of a midpoint 3.

Vertical Angles Theorem 4.

SAS Congruence Postulate 5.

Corres. parts of ≅ ∆’s are ≅ 6.

Alternate Interior Angles Converse T

Example 3: Using more than one pair of triangles

Given:  1 ≅  2,  3≅  4 Prove ∆BCE≅∆DCE Plan for proof: information you have about ∆BCE and ∆DCE is that and that CE ≅CE.

The only  1 ≅  2 Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent congruent to get a third piece of information about ∆BCE and ∆DCE.

triangles are C 2 1 D B E 3 4 A

Given:



1

≅ 

2,



3

≅ 

4.

Prove ∆BCE

≅

∆DCE

Statements: 1.

 1≅  2,  3≅  4 2. AC ≅ AC 3. ∆ABC ≅ ∆ADC 4. BC ≅ DC 5. CE ≅ CE 6. ∆BCE≅∆DCE D C 1 2 4 3 A E B Reasons: 1. Given 2. Reflexive property of Congruence 3. ASA Congruence Postulate 4. Corres. parts of ≅ ∆’s are ≅ 5.

6.

Reflexive Property of Congruence SAS Congruence Postulate

Goal 2: Proving Constructions are Valid

In Lesson 3.5 you learned to copy an angle using a compass and a straight edge. The construction is summarized on pg. 159 and on pg. 231.

Using the construction summarized above, you can copy  CAB to form  FDE. Write a proof to verify the construction is valid.

Plan for Proof

Show that ∆CAB ≅ ∆FDE.

Then use the fact that corresponding parts of congruent congruent to conclude that  CAB ≅ construction, assume triangles  the FDE.

you are By can following statements: – AB ≅ DE – AC ≅ DF – BC ≅ EF Same compass setting is used Same compass setting is used Same compass setting is used D A E F B C

C

Given: AB

≅

DE, AC

≅

Prove



CAB

≅ 

FDE DF, BC

≅

EF

Statements: 1. AB ≅ DE 2. AC ≅ DF 3. BC ≅ EF 4. ∆CAB ≅ ∆FDE 5.

 CAB ≅  FDE A F B D E Reasons: 1. Given 2. Given 3. Given 4. SSS Congruence Post 5. Corres. parts of ≅ ∆’s are ≅.