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"To accomplish great things, we must dream as well as act." Anatole France Bellwork – Write a theorem about the following Justify your theorem Leg-Leg Congruence Leg Angle Congruence Hypotenuse-Angle Congruence Hypotenuse-Angle Congruence "To accomplish great things, we must dream as well as act." Anatole France Bellwork – Write a theorem about the following Justify your theorem Leg-Leg Congruence If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent Hypotenuseleg Congruence If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. Leg Angle Congruence If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. Hypotenuse-Angle Congruence If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle then the two triangles are congruent Chapter 4.6 Isosceles Triangles Objective: To understand and be able to use the properties of isosceles and equilateral triangles. Check.4.10 Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals, polygons, and solids). Spi.4.3 Identify, describe and/or apply the relationships and theorems involving different types of triangles, quadrilaterals and other polygons. What are the 5 ways to prove Triangles congruent a. SSS, Side Side Side b. SAS, Side Angle Side c. ASA, Angle Side Angle d. AAS, Angle Angle Side e. SSSAAA 3 sides, 3 angles • f. What does CPCTC mean? R S P Given: PQR, PQRQ Prove: P R Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. B If AB CB then A C Q Statement 1. 2. 3. 4. 5. 6. 7. Reason A C Let S be midpoint of PR 1. Every segment has one midpoint Draw a segment SQ 2. Two points determine a line If two angles of a triangle are congruent, then the sides opposite those angles PScongruent. RS 3. Midpoint are B Theorem QS QS 4. Reflexive Property If A C then AB CB PQRQ 5. Given PQS RQS 6. SSS P R 7. CPCTC A C Check.4.10 Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals, polygons, and solids). What about an equilateral triangle? • If ABBCCA • ABBC then A C B • BCCA then A B Each Angle of an Equilateral Triangle = 60 C A B A = 60 B 60 • =What C = 60 • • • • does A, B and C equal? A +B + C = 180 A 3 A = 180 A = 60, B = 60, C = 60 C K Find the missing measure G H If GHHK, HJJK and mGJK = 100 What is the measure of HGK? KHJ + HKJ + KJH = 180 KHJ = HKJ, set equal to x x+ x + 100 = 180 KHG + HGK + GKH = 180 2x = 180-100= 80 HGK = GKH, set to y x = 40 140 + 2y = 180 KHJ = HKJ, = 40 2y = 40 KHG + KHJ = 180 y = 20 KHG = 140 HGK = GKH = 20 J Find the missing measure EFG is equilateral and EH bisects E Find m1 and m2 E m1 + m2 = 60 m1 = m 2 – bisector m1 = 60/2 = 30 EFH + m1 + EHF = 180 60+ 30 + 15x = 180 15x = 90 x=6 1 2 15x F H G Find the missing measure EFG is equilateral and EH bisects E, EJ bisects 2 Find HEJ and EJH and EJG m1 + m2 = 60 m1 = m 2 – bisector m1 = 60/2 = 30 EFH + m1 + EHF = 180 60+ 30 + 15x = 180 15x = 90 x=6 E 1 2 15x F H J G HEJ = 1/2 m 2 – bisector HEJ = ½ (30) =15 HEJ + EJH + JHE = 180 15 + EJH + 90= 180 EJH = 75 EJG = 105 Practice Assignment • Page 287, 10 – 24 even • Honors; page 288 16 – 42 even