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GEOMETRY MODULE 1 LESSON 24 CONGRUENECE CRITERIA FOR TRIANGLES βASA and SSS OPENING EXERCISE Use the provide 30° angle as one base angle of an isosceles triangle. Use a compass and straight edge to construct an appropriate isosceles triangle around it. You may extend the base if you wish. 1. Strike an arc A with the compass needle at the vertex. Note the intersection points. 2. On the opposite end of the base segment, strike an arc B the same width as arc A. 3. Open your compass to the width of the two intersection points from arc A (the original arc). 4. With this width, strike an arc with the compass needle on one of the intersection points from arc A. 5. Strike the same arc from the intersection point of arc B. 6. Draw the segment to complete the isosceles triangle. Does using a given angle measure guarantee that all the triangles constructed in class have corresponding sides of equal lengths? No, side lengths may vary. MOD1 L24 1 DISCUSSION We introduce two new triangle congruence criteria. ο· Angle-Side-Angle Triangle Congruence Criteria (ASA): Given two triangles βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ². If πβ πΆπ΄π΅ = πβ πΆβ²π΄β²π΅β² (Angle), π΄π΅ = π΄βπ΅β (Side), and πβ πΆπ΅π΄ = πβ πΆβ²π΅β²π΄β² (Angle), then the triangles are congruent. ο· Side-Side-Side Triangle Congruence Criteria (SSS): Given two triangles βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ². If π΄π΅ = π΄βπ΅β (Side), π΄πΆ = π΄βπΆβ (Side), and π΅πΆ = π΅βπΆβ (Side), then the triangles are congruent. As we did with SAS, we can prove these two congruence criteria through basic rigid motions. ASA SSS NOTE: For ASA, the two angles given/found must be connected to the given/found side. MOD1 L24 2 PRACTICE Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them. 1. Given: M is the midpoint of Μ Μ Μ Μ π»π and πβ π» = πβ π. ASA πβ π» = πβ π is given. π»π = ππ by definition of midpoint πβ πΊππ» = πβ π ππ by vertical angles So, βπΊππ» β βπ ππ 2. Given: Rectangle JKLM with diagonal Μ Μ Μ Μ Μ πΎπ SSS/SAS/ASA πβ π½ = πβ πΎ = πβ πΏ = πβ π by definition of rectangle π½π = πΎπΏ and π½πΎ = ππΏ by definition of rectangle πΎπ = πΎπ by Reflexive So, βπ½πΎπ β βπΏππΎ 3. Given: π π = π π΅, π΄π = ππ SAS πβ π΄π π = πβ ππ π΅ by vertical angles βππ π΅ is an isosceles triangle. πβ π΄π΅π = πβ πππ΅ Base angles of an isosceles triangles are equal in measure. So, βπ΄π π β βππ π΅ and βπ΄π΅π β βπππ΅ MOD1 L24 3 1 1 4. Given: π΄π΅ = π΄πΆ, π΅π· = 4 π΄π΅, πΆπΈ = 4 π΄πΆ SAS βπ΅π΄πΆ is an isosceles triangle. π΅π· = πΆπΈ by Transitive Property π΄π· = π΄πΈ by Transitive Property thru work below ο· π΄π΅ β π΅π· = π΄π· Partition property (segment subtraction) ο· π΄πΆ β πΆπΈ = π΄πΈ Partition property (segment subtraction) So, βπ΄π΅πΈ β βπ΄πΆπ· 5. Given: Circles with centers A and B intersect at C and D Prove: β πΆπ΄π΅ β β π·π΄π΅ What congruence criteria would be best here? STEP JUSTIFICATION 1 Circles with centers A and B intersect at C and D Given 2 π΄πΆ = π΄π· Radius of Circle 3 π΅πΆ = π΅π· Radius of Circle 4 π΄π΅ = π΄π΅ Reflexive Property 5 βπ΄πΆπ΅ β βπ΄π·π΅ SSS Corresponding angles of congruent 6 β πΆπ΄π΅ β β π·π΄π΅ triangles are congruent. MOD1 L24 4 SUMMARY ο· Angle-Side-Angle Triangle Congruence Criteria (ASA): Given two triangles βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ². If πβ πΆπ΄π΅ = πβ πΆβ²π΄β²π΅β² (Angle), π΄π΅ = π΄βπ΅β (Side), and πβ πΆπ΅π΄ = πβ πΆβ²π΅β²π΄β² (Angle), then the triangles are congruent. ο· Side-Side-Side Triangle Congruence Criteria (SSS): Given two triangles βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ². If π΄π΅ = π΄βπ΅β (Side), π΄πΆ = π΄βπΆβ (Side), and π΅πΆ = π΅βπΆβ (Side), then the triangles are congruent. ο· The ASA and SSS criteria implies the existence of a congruence that maps one triangle onto the other. MOD1 L24 5