Warm Up: Tell whether it is possible to draw each triangle. 1.Acute scalene triangle 2.Obtuse equilateral triangle 3.Right isosceles triangle 4.Scalene equiangular triangle 5.Right scalene triangle

4.3 Congruent Triangles

4.3 – Congruent Triangles Two geometric figures are congruent if they have exactly the same size and shape. When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.

A B C P Q R Ex 1: Illustrate the two triangles. What angles corresponding to what angles?

Ex 2: ABCD is  to HGFE, find x and y. A B DC F E G H 91 o 86 o 9cm (5y-12) o (4x-3)cm 113 o

Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Properties of Congruent Triangles Reflexive Property: Every triangle is congruent to itself. Symmetric Property of Congruent Triangles: If ABC =̃ DEF, then DEF =̃ ABC Transitive Property of Congruent Triangles: If ABC =̃ DEF and DEF =̃ JKL, then ABC =̃ JKL.

Ex 4: Given: seg RP  seg MN, seg PQ  seg NQ, seg RQ  seg MQ, m  P = 92 o and m  N is 92 o. Prove: ΔRQP  ΔMQN R P Q N M 92 o

Section 4.4 Proving Triangles Congruent: SSS and SAS

Using the SSS Postulate Side side side postulate: If three sides of one triangle are congruent to three sides of another triangle, the the two triangles are congruent.

Example 1 Prove the triangles congruent. A B C D StatementReason

Use the SSS Congruence Postulate to show that  ABC   FGH.

Using the SAS Postulate Side angle side postulate: If two sides and the included angle of one triangle are congruent to two sided and the included angle of another triangle, then the two triangles are congruent.

Example 2 Prove the triangles congruent. AB C D StatementReason

Section 4.5 Proving Triangles Congruent: ASA and AAS

Using the ASA Postulate Angle side angle postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the two triangles are congruent.

Example 1 Prove the triangles congruent. StatementReason M N O P M N O P N P

Using the AAS Theorem Angle angle side theorem: If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

Example 2 Prove the triangles congruent. StatementsReasons P Q R S T

Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC StatementsReasons