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Geometry Lesson 4.6 Isosceles, Equilateral, and Right Triangles Warm Up: Key Proof Concepts So far in Chapter 4 we have learned some key concepts about congruent figures: By definition: All corresponding sides and angles are congruent If figures , then corresp. sides and s If corresp. sides and s , then figures For triangles: It is not necessary to show ALL sides and angles are congruent 4 methods: SSS, SAS, ASA, AAS 1. CPC In this lesson we will go beyond just proving triangles are congruent We will prove other properties of a figure after we know that two triangles are congruent We can abbreviate the definition of congruence as follows: CPC: Corresponding Parts are Congruent If s , then CPC (converse: if CPC, then s ) Example 2a: Apply CPC Given: PQ RQ and PS RS Prove: PQS RQS Q Plan for proof: Know: PQ RQ and PS RS Can show: QS QS R P Logic: Show PQS RQS using SSS, then use CPC to show PQS RQS Statement PQ RQ and PS RS QS QS Reason S Given Reflexive Property of Congruence PQS RQS SSS PQS RQS CPC Example 2b: Apply CPC Given: AB || CD and BC || DA Prove: AB CD B C Plan for proof: Know: parallel sides A D Can show: CBD ADB and CDB ABD (alt int s) and BD BD (reflexive) Logic: Use ASA to show ADB CBD, then use CPC to show AB CD S. AB || CD and BC || DA CBD ADB BD BD ADB CBD AB CD Reason Given Alt. Int. Angles Reflexive Property of Congruence SAS CPC Warm Up: Review of Special Triangles Classify the triangles below Can you name the sides of the isosceles and right triangles? leg leg leg base isosceles equilateral right leg 1. Base Angles Theorem If two sides of a triangle are congruent, then their opposite angles are congruent B B and C are the base angles of the A isosceles ABC C If two angles of a triangle are congruent, then their opposite sides are congruent 2. Corollaries to Base Angle Theorem A If a triangle is equilateral, then it is equiangular If a triangle is equiangular, then it is equilateral equilateral B (biconditional) ↔ equiangular C Example 1 State a reason that allows you to solve for x (a) (b) Base Angles Theorem 3x = 24 x=8 Base Angles Theorem (c) if equiangular, then equilateral 3x – 11 = 2x + 11 x = 22 2x – 1 = x + 3 x=4 Practice 1 State a reason that allows you to solve for x & y (a) (b) hint: find the angle measures first Corollary to Base Angle Theorem 3x= 45 X=15 y+7=45 Y= 38 Corollary to the base angle theorem 3x+3x+3x=180 9x=180 X=20 Example 2 Find the values of x and y: F Finding x: EFG is equilateral G x° 120° y° x° H x° y° E If equilateral, then equiangular Finding y: GEH is isosceles (2 sides ) 3x = 180 x = 60 If 2 sides , then opp. s (Base Angle Theorem) mEGH = 180° - 60° = 120° 2y = 180 – 120 = 60 y = 30 3. Hypotenuse-Leg Theorem If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent A D B C E F If BC EF and AC DF, then ABC DEF Example 3 Given: AB AC and AM BC Prove: ABM ACM Plan for proof: Show s are right, AM AM, then use HL to show the triangles are Statement AB AC and AM BC AM AM AMB & AMC are rt. s Reason Given Reflexive Property Definition of Perpendicular Lines ABM & ACM are rt. s Definition of right triangles ABM ACM Hypotenuse-Leg Theorem Practice 3 Given: D is the midpoint of CE; BCD and FED are right angles; and BD FD F Prove: BCD FED B Statement C Reason D is midpoint of CE Given BCD and FED are rt s Given BD FD Given BCD and FED are rt s CD ED BCD FED D E Definition of right triangle Definition of midpoint Hypotenuse-Leg Theorem Assignment Ch 4.6 w/s