Proving Triangles Congruent Geometry D – Chapter 4.4.

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Presentation transcript:

Proving Triangles Congruent Geometry D – Chapter 4.4

SSS - Postulate If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)

Example #1 – SSS – Postulate Use the SSS Postulate to show the two triangles are congruent. Find the length of each side. AC = BC = AB = MO = NO = MN =

Definition – Included Angle K is the angle between JK and KL. It is called the included angle of sides JK and KL. What is the included angle for sides KL and JL? L

SAS - 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 triangles are congruent. (SAS) S A S S A S by SAS

Example #2 – SAS – Postulate Given: N is the midpoint of LW N is the midpoint of SK Prove: N is the midpoint of LW N is the midpoint of SK Given Definition of Midpoint Vertical Angles are congruent SAS Postulate

Definition – Included Side JK is the side between J and K. It is called the included side of angles J and K. What is the included side for angles K and L? KL

ASA - 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 triangles are congruent. (ASA) by ASA

Example #3 – ASA – Postulate Given: HA || KS Prove: HA || KS, Given Alt. Int. Angles are congruent Vertical Angles are congruent ASA Postulate

Note: is not SSS, SAS, or ASA. Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. by SSS by SAS

Example #4 – Paragraph Proof Given: Prove: is isosceles with vertex bisected by AH. Sides MA and AT are congruent by the definition of an isosceles triangle. Angle MAH is congruent to angle TAH by the definition of an angle bisector. Side AH is congruent to side AH by the reflexive property. Triangle MAH is congruent to triangle TAH by SAS. Side MH is congruent to side HT by CPCTC.

A line to one of two || lines is to the other line. Example #5 – Column Proof Given: Prove: Given has midpoint N Perpendicular lines intersect at 4 right angles. Substitution, Def of Congruent Angles Definition of Midpoint SAS CPCTC

Summary Triangles may be proved congruent by Side – Side – Side (SSS) Postulate Side – Angle – Side (SAS) Postulate, and Angle – Side – Angle (ASA) Postulate. Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).