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4-9 Isosceles and Equilateral Triangles Objectives Use CPCTC to prove parts of triangles are congruent. Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Vocabulary CPCTC legs of an isosceles triangle vertex angle base base angles Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles CPCTC an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Remember! SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent CPCTC uses congruent triangles to prove corresponding parts congruent Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Example: Engineering Application A and B are on the edges of a ravine. What is AB? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent. AB = 18 mi. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Example: Proving Corresponding Parts Congruent Given: YW bisects XZ, XY YZ. Prove: XYW ZYW Z Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles ZW WY Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles YOU TRY Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles SOULTION PR bisects QPS and QRS Given RP PR Reflex. Prop. of ∆PQR ∆PSR ASA PQ PS CPCTC Holt McDougal Geometry QRP SRP QPR SPR Def. of bisector 4-9 Isosceles and Equilateral Triangles Example: Using CPCTC In the Coordinate Plane Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3) Prove: DEF GHI Step 1 Plot the points on a coordinate plane. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Step 2 Use the Distance Formula to find the lengths of the sides of each triangle. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Step 3 Write statements justifying the triangles are congruent and proving angles congruent. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Recall: an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base. The base angles are the two angles that have the base as a side. Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles B 1 and 2 are the base angles. 3 is the vertex angle. 𝑨𝑩 is the base. A 𝑩𝑪 and 𝑨𝑪 are legs. Holt McDougal Geometry C