4-6 Objective: Use Congruent Triangles to Prove Corresponding Parts Congruent

Reminder about Congruent Figures If two triangles are congruent, then all the corresponding parts are congruent. If we can say two triangles are congruent, then we can say certain parts (sides or angles) are congruent, even if we didn’t use those parts.

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

EXAMPLE 1 Use congruent triangles Explain how you can use the given information to prove that the hanglider parts are congruent. SOLUTION GIVEN 1 2,  RTQ RTS PROVE QT ST If you can show that QRT SRT, you will know that QT ST. First, copy the diagram and mark the given information.

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.

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 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. SOLUTION GIVEN 1 2, 3 4 PROVE BCE DCE 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 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. To prove that CB CD, you can first prove that 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 2. 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. 3. Using the information in the diagram at the right, write a plan to prove that PTU UQP.

GUIDED PRACTICE for Examples 2 and 3 Given TU PQ Given PT QU Reflexive property PU 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 STATEMENTS REASONS SSS PTU UQP By SSS

In Conclusion To show any two corresponding parts of a triangle are congruent show they are congruent using one of the postulates –Write a congruence statement –Make sure the parts are in the right places in order to show they’re congruent

Examples Go to number 3 on page 259

Homework 1, 3-11, 16 – 20 evens, 23, 24, 29, 30, 33 – 35, 37 Bonus: 27, 38, 40