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Congruent Triangles Geometry Chapter 4 4.1 Triangles and Angles Classification by Sides: Triangles and Angles Classification by Angles Parts of Triangles Interior angle Exterior angle Vertex angle leg leg hypotenuse leg Base angle Base angle base leg Theorems Involving Triangles The sum of the measures of the angles of a triangle = 180° The measure of the exterior angle of a triangle = the sum of the two remote interior angles. B C A 3 2 1 Corollaries to Triangle Theorems The acute angles of a right triangle are complementary. Each angle of an equiangular triangle has a measure of 60°. In a triangle, there can be at most one right angle or one obtuse angle. ¬ Examples Sides of lengths 2mm, 3mm and 5mm. Sides of lengths 3m, 3m, 3m. Sides of lengths 8m, 8m, 5m. Examples Angles of measures 90, 25, 65. Angles of measures 60, 60, 60. Angles of measures 80, 70, 30. Angles of measures 140, 30, 10. Examples A triangle has angles that measure x, 7x, and x. Find x. Examples A right triangle has angle measures of x and (2x-21). Find x. Examples Find the measure of the exterior angle shown. 4.2 Congruence and Triangles Congruent – same size, same shape Congruent Polygons(Triangles) – Two polygons (triangles) are congruent iff their corresponding sides and corresponding angles are congruent E B A If ΔABC then A D B E C F C D ΔDEF, AB DE BC EF AC DF F Theorems about Congruent Figures If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. S N R If R M and S then T O T N, M O Examples L 110° H M G (2x +3)m (7y + 9)° 87° O 72° 10m N E If LMNO EFGH, find x and y. F Examples 4.3-4.3 Proving Triangles Congruent SSS – Side Side Side – If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. If AB DE BC EF AC DF, then ABC DEF SAS SAS – Side Angle Side – If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. If AB DE BC EF B E, then ABC DEF ASA ASA – Angle Side Angle – If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. If A D C F AC DF, then ABC DEF AAS AAS – Angle Angle Side – If two angles and a non-included side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. If A D C F AB DE, then ABC DEF HL HL – Hypotenuse Leg – If the hypotenuse and leg of one RIGHT triangle are congruent to the hypotenuse and leg of another RIGHT triangle then the triangles are congruent. D A C B F If ABC,DEF Right s, AB DE, AC DF, then ABC DEF. E 4.5 Using Congruent Triangles Definition of Congruent Triangles (rewritten) Corresponding Parts of Congruent Triangles are Congruent CPCTC is used often in proofs involving congruent triangles. M A is the midpoint of MT. A is the midpoint of SR. MS ll TR 1. A is the midpoint of MT. A is the midpoint of SR. R A S 1. Given T UR ll ST R and T are right angles U R 1. UR ll ST 1. Given R and T are right angles T S 4.6 Isosceles, Equilateral and Right Triangles B If two sides of a triangle are congruent, then the angles opposite are congruent. (Base angles of an isosceles triangle are congruent. Converse – If two angles of a triangle are C A congruent, then the sides If BA BC, then A C. opposite are congruent. If A C, then BA BC. More Corollaries B A C If a triangle is equilateral, then it is equiangular. If a triangle is equiangular then it is equilateral. Examples Find x and y. y 35 x Examples Find the unknown measures. ? 50 ? Examples Find x. (x-11) in 33 in Examples Find x and y. y 40 x