4.4-4.5 & 5.2: Proving Triangles Congruent

4.4-4.5 & 5.2: Proving Triangles Congruent p. 206-221, 245-251 Adapted from: http://jwelker.lps.org/lessons/ppt/geod_4_4_congruent_triangles.ppt

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 = 5 BC = 7 AB = MO = 5 NO = 7 MN = By SSS

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: Statement Reason N is the midpoint of LWN is the midpoint of SK Given 1 1 Definition of Midpoint 2 2 3 Vertical Angles are congruent 3 SAS 4 4

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: Reasons Statement Given 1 HA || KS, 1 Alt. Int. Angles are congruent 2 2 Vertical Angles are congruent 3 3 4 ASA Postulate 4

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

Example Given: Prove: Statement Reason 1) Given 1)

A C B D F E AAS (Angle, Angle, Side) • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . . then the 2 triangles are CONGRUENT!

Example Given: Prove: Statement Reason 1) Given 1) 2) 2)

A C B D F E HL (Hypotenuse, Leg) ***** only used with right triangles**** • If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

Example Given: Prove: Statement Reason 1) Given 1) 2) 2)

FOR ALL TRIANGLES SSS ASA AAS SAS FOR RIGHT TRIANGLES ONLY HL LL HA LA The Triangle Congruence Postulates &Theorems Only this one is new

Summary • Any Triangle may be proved congruent by: (SSS) (SAS) (ASA) (AAS) • Right Triangles may also be proven congruent by HL ( Hypotenuse Leg) • Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).

A C B Example 1 D E F

A C B D E F Example 2 • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? No ! SSA doesn’t work

A C B D Example 3 • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? YES ! Use the reflexive side CB, and you have SSS

Name That Postulate (when possible) SAS ASA SSA SSS

Name That Postulate (when possible) AAA ASA SSA SAS

Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS

Let’s Practice ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS:

Homework Assignment

4.4-4.5 & 5.2: Proving Triangles Congruent