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CHAPTER 4 Congruent Triangles What does CONGRUENCE mean? • Congruent angles- have equal measures • Congruent segments- have equal lengths • What did you notice about these pictures? • What does it mean for two objects or figures to be congruent? • Congruent figures- have the same shape and size • What about congruent triangles? Congruent Triangles (same shape and size) Are these triangles congruent? Can you rotate or reflect them so that they fit on top of one another? What are the corresponding angles? C F B A E D What are the corresponding sides? C F B A E D • Congruent triangles have the same shape • Corresponding angles are congruent • Congruent triangles have the same size • Corresponding sides are congruent Definition of Congruent Triangles • Two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangle are congruent • When naming congruent triangles, name the corresponding vertices in the same order • Example: If ABC ≅ XYZ then name the corresponding parts (angles and sides) Class work • p.221 #1-7 Homework • p.222-223 #8-18 even, 20, 21, 22-28 even Congruent Polygons • Polygon- closed plane figure (lies in a plane) formed by • • • • three or more segments. Each segment intersects two other segments at their endpoints Congruent polygons- polygons that have congruent corresponding parts “Corresponding parts”-matching sides and angles Naming congruent polygons- list corresponding vertices in the same order Polygons are congruent if and only if their vertices can be matched up so that their corresponding parts are congruent. Congruent Figures • Congruent polygons – have congruent corresponding parts • List corresponding vertices in the same order • List all congruent corresponding parts (sides & angles) ABCD ≅ HGFE B A C F D E G H Third Angles Theorem • If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. • Proof of this Theorem on p.220 Example: F C B A E D • If ∠ = 40, ∠ = 60, • And ∠ = 40 ∠ = 60, • What can be said about ∠ ∠? Example • ∆ ≅ ∆ • If ∠ = 20 + 14 ∠ = 30 − 36, • Find the value of x and determine the measures of ∠ and ∠ •x=5 • ∠ = 114 ∠ = 114 Homework • p.222-223 #30, 31 35, 37, 39-41 Congruence and Triangles worksheet all Building Congruent Triangles Activity • Spaghetti noodles • Straws p.225 Activities 1 and 2 Triangle Congruence (4-2) Side-Side-Side (SSS) Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Triangle Congruence (4-2) Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle in a second triangle, then the two triangles are congruent. Problem 3 Examples p.229 • A-D Class work • p.230-231 #3-4, 11, 12, 13, 14, 24, 25, 26 Homework • SSS and SAS Congruence worksheet Triangle Congruence (4-3) Angle-Angle-Side (AAS) Theorem If two angles and a non-included side of one triangle are congruent to two angles and a non-included side in a second triangle, then the triangles are congruent. Triangle Congruence (4-3) Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side in a second triangle, then the two triangles are congruent. Class Work • p.238 #1-4, 6, 7-9, 16-18 Homework • Complete SSS/SAS Congruence worksheet (if not already done) • Complete SSS, SAS, ASA, and AAS Congruence worksheet #1-18 all Proofs using SSS and SAS • #8 on p.230 • Given: • Prove: Class work • Class tries #9, 10, 16, and 17 in groups • Present to the class Homework • p.232 #28 (SSS and SAS) • Quiz tomorrow • Naming corresponding parts of congruent triangles • State 4 ways to prove triangles are congruent • (SSS, SAS, ASA, AAS) • Can two triangles be proved congruent and if, so how? • Proof- fill in missing statements and reasons (4 steps) Proofs using ASA and AAS • Proof of AAS Theorem on p.236 • You cannot use AAS as a reason! • Given: • Prove: Class Work • Class tries p.238-239 #11-15 in groups • Present to the class Homework • p.239 #11, 19-20 (ASA and AAS)