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Triangle Congruency Classifying Triangles by Sides Equilateral Triangle Isosceles Triangle 3 congruent sides At least 2 congruent sides Scalene Triangle No congruent sides Classifying Triangles by Angles Acute Triangle 3 acute angles Equiangular Triangle Right Triangle Obtuse Triangle 3 congruent angles 1 right angle 1 obtuse angle Note: An equiangular triangle is also acute. Terms to remember B Vertex Plural: Vertices A C Vertex Vertex More terms B side opposite C A C adjacent sides Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. B A C mA + m B + m C = 180° Term congruent figures – Two geometric figures that have exactly the same size and shape. All pairs of corresponding angles and sides are congruent. symbol: Example 1 Identify all pairs of congruent corresponding parts and write a congruence statement A F, C D, B E AB FE, BC ED, CA DF ∆ABC ∆ FED M Example 3 T (2x+30)° Find the value of x. N 55° 65° L R mM = 180º - 55º - 65 º mM = 60 º mM = mT 60 = 2x + 30 x = 15 S Try This! Find the value of x. B 87° A 22° F C (4x+15)° D E Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Q M If Side Side Side MN QR, NP RS, and PM SQ, S then MNP QRS R N P T S Example 1 V M I C Given: CI MV, IS VT, SC TM Prove: CSI MTV Statements 1. CI MV, IS VT, SC TM 2. CSI MTV Reasons 1. Given 2. SSS Congruence Post. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. X If Side Angle Side PQ WX, Q X, and QS XY, then PQS WXY Q P Y W S C B Example 2 1 Prove: ∆AEB ∆CED E 2 D A Statements Reasons 1. AE CE, BE DE 1. Given 2. 1 2 2. Vertical Angles Theorem 3. AEB CED 3. SAS Congruence Post. Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. B E A C D F If A D, AC DF, and C F, then ABC DEF Angle-Angle-Side (AAS) Congruence Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. B E A C D F If A D, C F, and BC EF, then ABC DEF Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. A D B C E If BC EF and AC DF, then F ABC DEF. Example 2 Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. H E Q N G J F Yes! AAS M P No! Try This! Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. C E 1 2 4 D 3 Yes! ASA F G No! AAA is NOT a congruence postulate or theorem Homework Page 238 numbers 1-3, 13,14 Page 244 numbers 2, 4, 10, 12, 14, 15 Page 251 Numbers 2-14 Evens