1. 4.5 Using Congruent Triangles

2. 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.

3. 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. Then you can use the fact that corresponding parts of congruent triangles are congruent to conclude that PQS ≅ RQS.

4. 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.

5. 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.

6. Example 1: Paragraph Proof
Because AD ║CD, it follows from the Alternate Interior Angles Theorem that ABD ≅CDB. For the same reason, ADB ≅CBD because BC║DA. By the Reflexive property of Congruence, BD ≅ 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.

7. 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.

8. Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR.
Statements:
A is the midpoint of MT, A is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
Reasons:
Given
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
Corres. parts of ≅ ∆’s are ≅
Alternate Interior Angles Converse

9. Example 3: Using more than one pair of triangles
Given: 1≅2, 3≅4
Prove ∆BCE≅∆DCE
Plan for proof: The only information you have about ∆BCE and ∆DCE is that 1≅2 and that CE ≅CE. 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 triangles are congruent to get a third piece of information about ∆BCE and ∆DCE. 2 4 3 1

10. Given: 1≅2, 3≅4. Prove ∆BCE≅∆DCE
Statements:
1≅2, 3≅4
AC ≅ AC
∆ABC ≅ ∆ADC
BC ≅ DC
CE ≅ CE
∆BCE≅∆DCE
Reasons:
Given
Reflexive property of Congruence
ASA Congruence Postulate
Corres. parts of ≅ ∆’s are ≅
Reflexive Property of Congruence
SAS Congruence Postulate

11. 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.

12. Plan for Proof
Show that ∆CAB ≅ ∆FDE. Then use the fact that corresponding parts of congruent triangles are congruent to conclude that CAB ≅ FDE. By construction, you can assume the following statements:
AB ≅ DE Same compass setting is used
AC ≅ DF Same compass setting is used
BC ≅ EF Same compass setting is used

13. Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅FDE
Statements:
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Reasons:
Given
SSS Congruence Post
Corres. parts of ≅ ∆’s are ≅.