2.3: Exploring Congruent Triangles GSE’s M(G&M)–10–4 Applies the concepts of congruency by solving problems on or off a coordinate plane; or solves problems using congruency involving problems within mathematics or across disciplines or contexts.

Definitions Congruent triangles: Triangles that are the same size and the same shape. A B C D E F In the figure DEF  ABC Congruence Statement: tells us the order in which the sides and angles are congruent

If 2 triangles are congruent: The congruence statement tells us which parts of the 2 triangles are corresponding “match up”. Means 3 Angles: 3 Sides: ORDER IS VERY IMPORTANT

Meaning A  T, R  E, C  F AR  TE, RC  EF, AC  TF A R C T E Example A R C T E F In the figure TEF  ARC Meaning A  T, R  E, C  F AR  TE, RC  EF, AC  TF

Congruent Triangles A Z B C X Y Write the Congruence Statement Example 2 Congruent Triangles Write the Congruence Statement A Z B C X Y Example 3

Example 3 : Congruence Statement Finish the following congruence statement: ΔJKL  Δ_ _ _ N M L M J L K N

Definition of Congruent: Triangles (CPCTC) Two triangles are congruent if and only if their corresponding parts are congruent. (tells us when Triangles are congruent) Example: Are the 2 Triangles Congruent. If so write The congruence statement.

Ex. 2 Are these 2 triangles congruent? If so, write a congruence statement.

Reflexive Property Does the Triangle on the left have any of the same sides or angles as the triangle on the right?

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

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

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

the 2 triangles are CONGRUENT! 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, . . . F E D A C B then the 2 triangles are CONGRUENT!

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

The Triangle Congruence Postulates &Theorems AAS ASA SAS SSS FOR ALL TRIANGLES LA HA LL HL FOR RIGHT TRIANGLES ONLY 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).

Example 1 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? A C B F E D No ! SSA doesn’t work

Example 3 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D A C B 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 SAS SAS SSA SAS Vertical Angles Reflexive Property (when possible) Vertical Angles Reflexive Property SAS SAS Vertical Angles Reflexive Property SSA SAS

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

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