1. 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.

2. 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,

3. 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.

4. 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.

5. 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

6. 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.
SOLUTION

7. 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.
TU PQ