1. Use Congruent Triangles
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
Suppose that ∆XYZ ∆RST. Complete each statement.
1. XY ?
ANSWER RS 2. Z ?
ANSWER T
3. m S = m ?
ANSWER Y

3. Warm-Up
4. If A B, m A = (2x + 40)º, and m B = (3x – 10)º, find x.
ANSWER 50

4. Example 1
Explain how you can use the given information to prove that the hang glider parts are congruent.
∠ RTQ RTS
GIVEN: ,
PROVE:
QT ST
SOLUTION
If you can show that QRT  SRT, you will know that QT  ST.

5. Example 1
First, copy the diagram and mark the given information. Then add the information you can deduce. In this case, RQT and RST are supplementary to congruent angles, so RQT  RST. Also, RT  RT.
Mark given information.
Add deduced information.
Two angle pairs and a non-included side are congruent, so by the AAS Congruence Theorem, QRT  SRT Because corresponding parts of congruent triangles are congruent, QT  ST.

6. Guided Practice
Explain how you can prove that
A C.
Since BD  BD by the Reflexive Property, the triangles are congruent by SSS. So, A  C because they are corresponding parts of congruent triangles.
ANSWER

7. Example 2
Surveying
Use the following method to find the distance across a river, from point N to point P.
Place a stake at K on the
near side so that NK NP
Find M, the midpoint of NK .
Locate the point L so that NK KL and L, P, and M
are collinear.
Explain how this plan allows you to find the distance.

8. Example 2
SOLUTION
Because NK NP and NK KL, N and K are congruent right angles. Because M is the midpoint of NK , NM  KM .
The vertical angles KML and NMP are congruent. So, MLK  MPN by the ASA Congruence Postulate. Then, because corresponding parts of congruent triangles are congruent, KL  NP. So, you can find the distance NP across the river by measuring KL.

9. Example 3
Use the given information to write a plan for proof.
GIVEN: ,
PROVE:
BCD DCE
SOLUTION
In BCE and DCE, you know 1  2 and CE  CE. If you can show that CB  CD , you can use the SAS Congruence Postulate.

10. Example 3
To prove that CB CD , you can first prove that
CBA CDA. You are given and CA CA by the Reflexive Property. You can use the ASA Congruence Postulate to prove that CBA CDA.
Plan for Proof
Use the ASA Congruence Postulate to prove that
CBA CDA. Then state that CB CD . Use the SAS Congruence Postulate to prove that BCE DCE.

11. Guided Practice
In Example 2, does it matter how far from point N you place a stake at point K ? Explain.
ANSWER
No, since M is the midpoint of NK, NM  MK. No matter how far apart the stakes at K and M are placed, the triangles will be congruent by ASA.

12. Guided Practice
Using the information in the diagram at the right, write a plan to prove that PTU UQP.
ANSWER
Since you already know that TU  QP and UP  PU, you need only show PT  UQ to prove the triangles are congruent by SSS. This can be done by showing right triangles QSP and TRU are congruent by HL leading to right triangles USQ and PRT being congruent by HL which gives you PT  UQ.

13. Example 4
Write a proof to verify that the construction for copying an angle is valid.
SOLUTION
Add BC and EF to the diagram. In the construction, AB , DE , AC , and DF are all determined by the same compass setting, as are BC and EF . So, you can assume the following as given statements.
GIVEN:
AB DE, AC DF, BC EF
PROVE:
D  A

14. Example 4
Plan For Proof
Show that CAB  FDE, so you can conclude that the corresponding parts A and D are congruent.
Plan in Action
STATEMENTS
REASONS
AB DE,
AC DF, BC EF
Given
FDE CAB
SSS Congruence Postulate
D A
Corresp. parts of
are .

15. Guided Practice
Look back at the construction of an angle bisector in Explore 4 on page 34. What segments can you assume are congruent?
AC and AB
ANSWER

16. Lesson Quiz
Tell which triangles you can show are congruent in order to prove AE = DE. What postulate or theorem would you use? 1.
ANSWER
AEC DEB by the AAS Cong. Thm. or by the ASA Cong. Post.

17. Lesson Quiz Write a plan to prove 1 2. 2. ANSWER
Show LM LM by the Refl. Prop. of Segs. Hence OLM NML by the SAS Cong. Post. This gives NLM OML, since Corr. Parts of are So by the Vert. Thm. and properties of s