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EXAMPLE 2

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. EXAMPLE 2.

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EXAMPLE 2

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  1. Use the following method to find the distance across a river, from point Nto point P. • Place a stake at Kon the near side so that NK NP • Find M,the midpoint of NK . • Locate the point Lso that NKKLand L, P,and M are collinear. EXAMPLE 2 Use congruent triangles for measurement Surveying

  2. Because NK NPand NK KL, Nand Kare congruent right angles. Because Mis the midpoint of NK, NMKM. The vertical angles KMLand NMP are congruent. So, MLK MPNby the ASA Congruence Postulate. Then, because corresponding parts of congruent triangles are congruent, KL NP. So, you can find the distance NPacross the river by measuring KL. EXAMPLE 2 Use congruent triangles for measurement • Explain how this plan allows you to find the distance. SOLUTION

  3. 1 2,3 4 GIVEN BCDDCE PROVE In BCEand 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 Use the given information to write a plan for proof. SOLUTION

  4. Use the ASA Congruence Postulate to prove that CBACDA.Then state that CB CD. Use the SAS Congruence Postulate to prove that BCE DCE. EXAMPLE 3 Plan a proof involving pairs of triangles To prove that CBCD, you can first prove that CBACDA. You are given 12 and 34. CACAby the Reflexive Property. You can use the ASA Congruence Postulate to prove that CBACDA. Plan for Proof

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

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

  7. STATEMENTS REASONS TU PQ Given PT QU Given Reflexive property PU PU PTUUQP SSS PTUUQP 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 for Examples 2 and 3 GUIDED PRACTICE

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