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.

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,

EXAMPLE 3 Plan a proof involving pairs of triangles Use the given information to write a plan for proof. GIVEN 1 2, 3 4 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.

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 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

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

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