1. Suppose that ∆XYZ ∆RST. Complete each statement.?
ANSWER RS 2. Z ?
ANSWER T

2. Suppose that ∆XYZ ∆RST. Complete each statement.
3. m S = m ?
ANSWER Y
4. If A B, m A = (2x + 40)º, and m B = (3x – 10)º, find x.
ANSWER 50

3. EXAMPLE 1
Use congruent triangles
Explain how you can use the given information to prove that the hanglider parts are congruent.
GIVEN ,
∠ RTQ RTS
PROVE
QT ST
SOLUTION
If you can show that QRT SRT, you will know that QT ST.
First, copy the diagram and mark the given
information.

4. Use congruent triangles
EXAMPLE 1
Use congruent triangles
Then add the information that 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, Because corresponding parts of congruent triangles are congruent,
QRT SRT
QT ST.

5. GUIDED PRACTICE
for Example 1
Explain how you can prove that
A C.
SOLUTION
Given
AB BC
Given
AD DC
Reflexive property
BD BD
ABD BCD
Thus the triangle by SSS
ANSWER

6. EXAMPLE 2
Use congruent triangles for measurement
Use the following method to find the distance across a river, from point N to point P.
Surveying
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.

7. EXAMPLE 2
Use congruent triangles for measurement
Explain how this plan allows you to find the distance.
SOLUTION
Because NK NP and NK KL , N and K are congruent right angles.
Then, because corresponding parts of congruent triangles are congruent, KL NP . So, you can find the distance NP across the river by measuring KL .
MLK MPN by the ASA Congruence Postulate.
Because M is the midpoint of NK , NM KM . The vertical angles KML and NMP are congruent. So,

8. EXAMPLE 3
Plan a proof involving pairs of triangles
Use the given information to write a plan for proof.
GIVEN ,
PROVE
BCD DCE
SOLUTION
In BCE and DCE, you know and CE CE . If you can show that CB CD , you can use the SAS Congruence Postulate.

9. EXAMPLE 3
Plan a proof involving pairs of triangles
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.

10. GUIDED PRACTICE for Examples 2 and 3
In Example 2, does it matter how far from point N you place a stake at point K ? Explain.
SOLUTION
No, it does not matter how far from point N you place a stake at point K . Because M is the midpoint of NK
Given
NM MK
Definition of right triangle
MNP MKL are
both right triangles
Vertical angle
KLM NMP
ASA congruence
MKL MNP

11. GUIDED PRACTICE
for Examples 2 and 3
No matter how far apart the strikes at K and M are placed the triangles will be congruent by ASA.
Using the information in the diagram at the right, write a plan to prove that PTU UQP.

12. GUIDED PRACTICE for Examples 2 and 3 STATEMENTS REASONS TU PQ PT QU
Given
TU PQ
Given
PT QU
Reflexive property
PU PU
SSS
PTU UQP
PTU UQP
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
Prove a construction
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. SSS Congruence Postulate FDE CAB
EXAMPLE 4
Prove a construction
Plan For Proof
Show that CAB FDE, so you can conclude that the corresponding parts A and D are congruent.
STATEMENTS
REASONS
AB DE
Given
AC DF, BC EF
Plan in Action
SSS Congruence Postulate
FDE CAB
Corresp. parts of
D A
are .

15. GUIDED PRACTICE
for Example 4
Look back at the construction of an angle bisector in Explore 4 on page 34. What segments can you assume are congruent?
SOLUTION
AC and AB

16. Daily Homework 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. Daily Homework 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