2.3: Exploring Congruent Triangles

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.

C F B D E A Definitions Congruent triangles: Triangles that are the ________ and the __________. 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 ISVERY IMPORTANT

C F R T E A In the figure TEF  ARC Example Meaning

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 ______________________ of sides JK and KL. What is the included angle for sides KL and JL?

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 ______________ of angles J and K. What is the included side for angles K and L?

ASA - Postulate If two angles and the included side of one triangle are _________ to two angles and the included side of a second triangle, then the triangles are __________. (ASA) by ASA __________

Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. by SSS by SAS

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

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

FOR ALL TRIANGLES FOR RIGHT TRIANGLES ONLY 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?

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?

Name That Postulate (when possible)

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

Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:

Homework Assignment

Assignment

2.3: Exploring Congruent Triangles