Using triangle congruence.

USING TRIANGLE CONGRUENCEavery common way of showing that two segments are congruent is by looking them as corresponding angles of congruent triangles

2Side-Angle Relations in a Triangleconsider an isosceles triangle POM thatOPOM OP N M

Side-Angle Relations in a TriangleFree powerpoint template: www.brainybetty.com3TheoremIsosceles Triangle Theorem if two sides of a triangle are congruent, then the angles opposite those side are congruent

Side-Angle Relations in a TriangleProve that an equilateral triangle ABC is also equiangularStatement ReasonAB BC definition of an equilateral triangle A C Isosceles triangle theoremAB AC Definition of an equilateral triangle B C isosceles triangle theorem A B C Transitive property∆ABC is equiangular. Def. of an equiangular angleFree powerpoint template: www.brainybetty.com4

Side-Angle Relations in a TriangleTheoremConverse of Isosceles Triangle Theorem if two angles of a triangle are congruent, then the sides opposite those angles are congruentFree powerpoint template: www.brainybetty.com5

Side-Angle Relations in a TriangleStatement Reason A C definition of an AB equiangular angleAC BC Converse ofAB BC Isosceles Triangle TheoremFree powerpoint template: www.brainybetty.com6

Side-Angle Relations in a TriangleStatement ReasonAB BC AC Transitive property∆ABC is equilateral Definition of an equilateral triangleFree powerpoint template: www.brainybetty.com7

Inequalities in a TriangleIs m C > m B ?Actual measurements shows that the statement is true, but there is a need to reason out why this is so A B CFree powerpoint template: www.brainybetty.com8

Inequalities in a TriangleExtendACto a point D such thatABADWe now have an isosceles triangle.Free powerpoint template: www.brainybetty.com9

Inequalities in a TriangleTheoremIf two sides of a triangle are not congruent, then the angles opposite these two sides are not congruent, and the larger angle is opposite the longer sideFree powerpoint template: www.brainybetty.com10

Inequalities in a TriangleIn ∆RSP RS = 35 RP = 31 PS = 52Which is the largest scale?Which is the smallest angle?Solution:The largest angle is R The largest angle is SFree powerpoint template: www.brainybetty.com11

Inequalities in a TriangleTheorem if two angles of a triangle are not congruent, then the sides opposite these two angles are not congruent, and the longer side is opposite the larger angleFree powerpoint template: www.brainybetty.com12

Inequalities in a TriangleIn the accompanying figure, OP = OM, m OPQ =145, and m POM =110What is the longest side of ∆OPM ?Solution:Since the POMIs the largest angleSo PM is the longest side of ∆OPMFree powerpoint template: www.brainybetty.com13

Congruence of right trianglesFree powerpoint template: www.brainybetty.com14

Congruence of right trianglesIn any right triangle,The side oppositeof the right triangle is called the hypotenuseThe two others are LegsFree powerpoint template: www.brainybetty.com15

Congruence of right trianglesTheoremLL congruence theorem if the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruentFree powerpoint template: www.brainybetty.com16

Congruence of right trianglesSince all right triangles are congruent then R M. thus, by ASA Congruence Postulate,We have ∆ORS ∆LMNFree powerpoint template: www.brainybetty.com17

Congruence of right trianglesTheoremLA congruence Theorem if a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent.Free powerpoint template: www.brainybetty.com18

Congruence of right trianglesSince all right angles are congruent, then T Y. Thus, by SAA congruence postulate, we have∆STU ∆XYZFree powerpoint template: www.brainybetty.com19

Congruence of right trianglesTheoremHyACongruence Theorem If an acute angle and the hypotenuse of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent.Free powerpoint template: www.brainybetty.com20

HyACongruence TheoremFree powerpoint template: www.brainybetty.com21

Congruence of right trianglesTheoremHyLCongruence theoremif a leg and the hypotenuse of one right triangle are congruent to a corresponding leg and the hypotenuse of another right triangle, then the triangles are congruent.Free powerpoint template: www.brainybetty.com22

Congruence of right trianglesFree powerpoint template: www.brainybetty.com23Extend ray DE to a point G such that GEAB. By SAS congruence Postulate, we have ∆ ABC ∆GEF. we get AC GF

Key ExpressionsFree powerpoint template: www.brainybetty.com24

Using triangle congruence.

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Using triangle congruence.