# EXAMPLE 2 Use congruent triangles for measurement Surveying EXAMPLE 2 Use congruent triangles for measurement 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 are collinear. KL and L, P, and M 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. 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 . EXAMPLE 3 Plan a proof involving pairs of triangles Use the given information to write a plan for proof. GIVEN

1 PROVE 2, BCD 3 4 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. EXAMPLE 3 Plan a proof involving pairs of triangles

To prove that CB CD , you can first prove that CBA CDA. You are given 1 2 and 3 4. 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. GUIDED PRACTICE 2. 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 NM MK MNP MKL are both right triangles KLM NMP MKL MNP Given Definition of right triangle Vertical angle ASA congruence 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.

3. Using the information in the diagram at the right, write a plan to prove that PTU UQP. SOLUTION for Examples 2 and 3 GUIDED PRACTICE STATEMENTS TU PQ Given PT QU Given

PU PU Reflexive property PTU PTU REASONS UQP SSS UQP By SSS This can be done by showing right triangles QSP and TRU are congruent by HL leading to right triangles TU gives PQ you USQ and PRT being congruent by HL which PT = UQ.