2.2a: Exploring Congruent Triangles CCSS G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 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. Congruent triangles: Triangles that are the same size and the same shape. C A F B D In the figure E 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”. ABC DEF Means 3 Angles: A D, B E, C F and 3 Sides: AB DE, BC EF, andAC DF ORDER IS VERY IMPORTANT Example C A In the figure R F T TEF E ARC A T, R E, C F AR TE, RC EF, AC TF Example 2 Congruent Triangles A Z B C X Example 3 JKL RST Write the Congruence Statement ABC ZXY Y R J ______ K S ______ ST KL ______ Example 3 : Congruence Statement Finish the following congruence statement: ΔJKL Δ_N M _ _L M J L K N Definition of Congruent: Two triangles are congruent Triangles if and only if their (CPCTC) corresponding parts are congruent. (tells us when Triangles are congruent) 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 2 2 AB = 5 7 74 MO = 5 NO = 7 MN = 52 72 74 ACB MON By SSS Definition – Included Angle J K is the angle between JK and KL. It is called the included angle of sides JK and KL. K L J What is the included angle for sides KL and JL? L K 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 L Q P A S A J S S K JKL PQR by SAS R Definition – Included Side J JK is the side between J and K. It is called the included side of angles J and K. K L J What is the included side for angles K and L? KL K L 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) J X Y K L JKL ZXY by ASA Z Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. A J R B C H I S K M O L P VABC VSTR by SSS VPNO VVUW by SAS N V T U W Note: VJHI is not SSS, SAS, or ASA. AAS (Angle, Angle, Side) • If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding non- C included side of another triangle, . . . then the 2 triangles are CONGRUENT! A D B F E 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 D then the 2 triangles are CONGRUENT! F E The Triangle Congruence Postulates &Theorems FOR ALL TRIANGLES SSS SAS ASA AAS FOR RIGHT TRIANGLES ONLY HL Only this one is new LL HA LA 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 Given the informatio n in the diagram, A is there any way to determine if CB DF ? C B D YES!! CAB DEF by SAS so CB DF by CPCTC E F Example 2 A C B D E F • 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 Example 3 A C D • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? B YES ! Use the reflexive side CB, and you have SSS Name That Postulate (when possible) SAS SSA ASA SSS Name That Postulate (when possible) AAA SAS ASA SSA Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AC FE For AAS: A F Homework Assignment