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"It's okay to make mistakes. Mistakes are our teachers -- they help us to learn." John Bradshaw Copy these into your Glossary Theorem Example Angle Sum Theorem The sum of the measures of the angles of a triangle is 180 Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, the third angles of the triangles are congruent. Exterior Angle Theorem Corollaries X mW + mX +mY = 180 If A F and B D, then C E, then The measure of an exterior mYZP = angle of a triangle is equal to mX +mY the sum of the measures of the two remote interior angles. The acute angles of a right triangle are complementary There can be at most one right or obtuse angel in a triangle Y W F A B D C E Y P X mG + mJ = 90 Acute S G H J Chapter 4.2 Angles of Triangles: Objective: Understand and apply the angle sum and exterior angle theorems. Check.4.11 Use the triangle inequality theorems (e.g., Exterior Angle Inequality Theorem, Hinge Theorem, SSS Inequality Theorem, Triangle Inequality Theorem) to solve problems. Check.4.12 Apply the Angle Sum Theorem for polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve contextual problems. Spi.4.11 Use basic theorems about similar and congruent triangles to solve problems. Angles of Triangle Cut out a triangle (1/2 size of a piece of paper) Label vertices A, B, and C (on front and back) Fold vertex B so it touches AC the fold line is parallel AC Fold A and C so they meet vertex B What do you notice about the sum of angles A, B and C? Tear of vertex A, and B Arrange A and B so they fill in the angle adjacent and supplementary to C. What do you notice about the relationship A and B and the angle outside C? "It's okay to make mistakes. Mistakes are our teachers -- they help us to learn." John Bradshaw Demonstrated 2 Theorems Theorem Example Angle Sum Theorem The sum of the measures of the angles of a triangle is 180 Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, the third angles of the triangles are congruent. Exterior Angle Theorem Corollaries X mW + mX +mY = 180 If A F and B D, then C E, then The measure of an exterior mYZP = angle of a triangle is equal to mX +mY the sum of the measures of the two remote interior angles. The acute angles of a right triangle are complementary There can be at most one right or obtuse angel in a triangle Y W F A B D C E Y P X mG + mJ = 90 Acute S G H J Angle Sum Theorem X Given ABC Prove: mA+mB+mC = 180 Statement ABC Line XY through A || CB 1 and CAY form a linear pair 1 and CAY are supplementary 5. m1+mCAY=180 6. mCAY= m2+m3 7. m1+m 2+m3=180 8. 1 C, 3 B 9. m1=mC, m3=mB 10. mC+m 2+mB=180 1. 2. 3. 4. A 1 2 C Y 3 B Reasons 1. 2. 3. 4. Given Parallel Postulate Def of linear pair If 2 ’s form a linear pair, they are supplementary 5. Def of supplementary ’s 6. Angle Addition Postulate 7. Substitution 8. Alternate Interior Angle Theorem 9. Def of congruent angles 10. Substitution m1 + 28 + 82 = 180 Find the missing Angles m1 + 110 = 180 m1 = 70 82 m1 = m2 vertical angles 28 1 m2 + m3 + 68 = 180 70+ m3 + 68 = 180 m3 + 138 = 180 m3 = 42 2 68 3 m1 + 74 + 43 = 180 Find the missing Angles m1 + 117 = 180 m1 = 63 79 43 1 m1 = m2 vertical angles 2 74 3 m2 + m3 + 79 = 180 63 + m3 + 79 = 180 m3 + 142 = 180 m3 = 38 Find the angle measures 3 2 50 78 1 120 4 m1 = 50 + 78, exterior angle theorem m1 = 128 m1 + m2 = 180, linear pair are supplemental 128 + m2 = 180 m2 = 52 m2 + m3 = 120 exterior angle theorem 52+ m3 = 120 m3 = 68 120 + m4 = 180, linear pair are supplemental m4 = 60 m4 + 56 = m5 exterior angle theorem 60+ 56 = m5 116= m5 5 56 Find the angle measures 5 4 3 32 41 64 38 2 1 m1 = 32 + 38 m1 = 70 m1 + m2 = 180, linear pair are supplemental m2 = 110 m2 = m3 +64 exterior angle theorem m3= 110 – 64 = 46 m3 + m4 +32 = 180 46 + m4 + 32 = 180 m4 = 102 m4 + m5 +41 = 180 102 + m5 +41 = 100 37= m5 29 Right Triangle m1 = 90 – 27 m1 = 63 27 1 Practice Assignment • Standard - page 248, 12 -32 Even • Honors - Page 189 24 – 44 Even