How to use triangle congruence and CPCTC to prove that parts of two triangles are congruent, and HL to prove two triangles congruent. Chapter 4.4 and 4.6GeometryStandard/Goal: 4.1

1. Check and discuss assignment from Thursday. 2. Work on Quiz Read, write, and discuss how to use triangle congruence and CPCTC to prove that parts of two triangles are congruent. 4. Read, write, and discuss how to prove triangles congruent using the HL Theorem. 5. Work on assignment.

CPCTC Corresponding parts of congruent triangles are congruent.

hypotenuse In a right triangle, the side opposite the right angle and is the longest side. legs The other two sides of a right triangle. hypotenuse leg

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, Then the triangles are congruent. B A C E D F

What other congruence statements can you prove from the diagram, in which SL SR, and  1  2 are given? SC SC by the Reflexive Property of Congruence, and LSC RSC by SAS.  3  4 by corresponding parts of congruent triangles are congruent. When two triangles are congruent, you can form congruence statements about three pairs of corresponding angles and three pairs of corresponding sides. List the congruence statements. Lesson 4-4

(continued) Lesson 4-4 SL SR Given SC SC Reflexive Property of Congruence CL CR Other congruence statement Sides:  1  2Given  3  4Corresponding Parts of Congruent Triangles  CLS  CRS Other congruence statement Angles: In the proof, three congruence statements are used, and one congruence statement is proven. That leaves two congruence statements remaining that also can be proved:  CLS  CRS CL CR

The Given states that  DEG and  DEF are right angles. What conditions must hold for that to be true?  DEG and  DEF are the angles the officer makes with the ground. So the officer must stand perpendicular to the ground, and the ground must be level. Lesson 4-4

One student wrote “ CPA MPA by SAS” for the diagram below. Is the student correct? Explain. There are two pairs of congruent sides and one pair of congruent angles, but the congruent angles are not included between the corresponding congruent sides. The triangles are not congruent by the SAS Postulate, but they are congruent by the HL Theorem. Lesson 4-6 The diagram shows the following congruent parts. CA MA  CPA  MPA PA

XYZ is isosceles. From vertex X, a perpendicular is drawn to YZ, intersecting YZ at point M. Explain why XMY XMZ. Lesson 4-6

Write a two–column proof. Given :  ABC and  DCB are right angles, AC DB Prove : ABC DCB StatementsReasons Lesson  ABC and  DCB are1. Given right angles. 2. ABC and DCB are2. Definition of a right triangle right triangles. 3. AC DB 3. Given 4. BC CB 4. Reflexive Property of Congruence 5. ABC DCB 5. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. (HL Theorem).

Kennedy, D., Charles, R., Hall, B., Bass, L., Johnson, A. (2009) Geometry Prentice Hall Mathematics. Power Point made by: Robert Orloski Jerome High School.