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Lesson 4.1/1-9 and 4.2/1-10

Topic: U7 L3 Triangle Sum Properties & Properties of Isosceles Triangles. EQ: What are the triangle sum properties and how can I use them to find the measures of angles in triangles?. Lesson 4.1/1-9 and 4.2/1-10. Classification By Sides. Classification By Angles. Classifying Triangles.

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Lesson 4.1/1-9 and 4.2/1-10

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  1. Topic: U7 L3Triangle Sum Properties & Properties of Isosceles Triangles EQ: What are the triangle sum properties and how can I use them to find the measures of angles in triangles? Lesson 4.1/1-9 and 4.2/1-10

  2. Classification By Sides Classification By Angles

  3. Classifying Triangles In classifying triangles, be as specific as possible. Obtuse, Isosceles Acute, Scalene

  4. 1 3 2 Triangle Sum Theorem **NEW The sum of the measures of the interior angles of a triangle is 180o. m<1 + m<2 + m<3 = 180°

  5. Property of triangles The sum of all the angles equals 180º degrees. 60º 90º 30º + 60º 180º 30º 90º

  6. Property of triangles The sum of all the angles equals 180º degrees. 60º 60º 60º 60º + 180º 60º 60º

  7. What is the missing angle? 70º 70º ? ? + 180º 70º 70º 180 – 140 = 40˚

  8. What is the missing angle? 90º ? 30º ? + 180º 90º 30º 180 – 120 = 60˚

  9. What is the missing angle? ? 60º 60º ? + 60º 60º 180º 180 – 120 = 60˚

  10. What is the missing angle? ? 30º 78º ? + 78º 30º 180º 180 – 108 = 72˚

  11. Find all the angle measures 35x 45x 10x 180 = 35x + 45x + 10x 180 = 90x 2 = x 90°, 70°, 20°

  12. What can we find out? The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building? 180 = 75 + 90 + x 180 = 165 + x 15˚ = x

  13. Corollary to Triangle Sum Theorem Acorollaryis a statement that readily follows from a theorem. The acute angles of a right triangle are complementary. m∠A + m∠B = 90o

  14. Find the missing angles. The tiled staircase shown below forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other angle. Find the measure of each acute angle. Con’t

  15. Find the missing angles. SOLUTION: 2x + x = 90 3x = 90 x = 30˚ 2x = 60˚

  16. Find the missing angles. 2x + (x – 6) = 90˚ 2x = 2(32) = 64˚ 3x – 6 = 90 3x = 96 (x – 6) = 32 – 6 = 26˚ x = 32

  17. Isosceles Triangleat least two sides have the same length 5 m 5 m 5 m 9 in 9 in 3 miles 3 miles 4 miles 4 in

  18. Properties of an Isosceles Triangle • Has at least 2 equal sides • Has 2 equal angles • Has 1 line of symmetry

  19. The vertexisthe angle between the two congruent sides. Parts of an Isosceles Triangle:

  20. The base anglesare the angles oppositethe congruent sides. Parts of an Isosceles Triangle:

  21. The baseis the side opposite the vertex angle Parts of an Isosceles Triangle:

  22. Isosceles Triangle ConjectureIf a triangle is isosceles, thenbase angles are congruent. Ifthen

  23. Ifthen Converse of Isosceles Triangle ConjectureIf a triangle has two congruent angles, then it is an isosceles triangle.

  24. Equilateral TriangleTrianglewith all three sides are congruent 7 ft 7 ft 7 ft

  25. Equilateral Triangle Conjecture An equilateral triangle is equiangular, and an equiangular triangle is equilateral.

  26. <68° and < a are base angles  they are congruent b Find the missing angle measures. ma = 68˚ Triangle sum to find <b m<b = 180 – 68 - 68 m<b = 180 -136 mb = 44˚ 68˚ a

  27. Find the missing angle measures. <c & <d are base angles and are congruent Triangle sum = 180° 180 = 119 + c + d 180 – 119 = c + d 61 = c + d <c = ½ (61) = 30.5 <d = ½ (61) = 30.5 mc = md = 30.5˚ 119˚ 30.5˚ c d

  28. Find the missing angle measures. E EFG is an equilateral triangle <E = <F = <G 180 /3 = 60 mE = mF = mG = 60˚ 60˚ G F 60˚

  29. Find the missing angle measures. Find mG. ∆GHJ is isosceles < G = < J x + 44 = 3x 44 = 2x x= 22 Thus m<G = 22 + 44 = 66° And m<J = 3(22) = 66°

  30. Find the missing angle measures. Find mN Base angles are = 6y = 8y – 16 -2y = -16 y= 8 Thus m<N = 6(8) =48°. m<P = 8(8) – 16 = 48°

  31. Find the missing angle measures. Using Properties of Equilateral Triangles Find the value of x. ∆LKM is equilateral m<K = m<L = m<M 180/3 = 60° 2x + 32 = 60 2x = 37 x = 18.5°

  32. Find the missing side measures. Using Properties of Equilateral Triangles Find the value of y. ∆NPO is equiangular ∆NPO is also equilateral. ft ft 5y – 6 = 4y +12 y – 6 = 12 y = 18 Side NO = 5(18) – 6 = 90ft

  33. Find the missing angle measures. Using the symbols describing shapes answer the following questions: b 45o d a c 36o Equilateral triangle all angles are equal Isosceles triangle Two angles are equal Right-angled triangle c = 180 ÷ 3 = 60o a = 36o d = 180 – (45 + 90) = 45o b = 180 – (2 × 36) = 108o

  34. Find the missing angle measures. A D B C A B C Equilateral triangle e = f = g = 60o c = d a = 64o c + d = 180 - 72 b = 180 – (2 ×64o ) = 52o h = i D c + d = 108 h + i = 180 - 90 c = d = 54o h + i = 90 h = i = 45o

  35. p = 50o q = 180 – (2 ×50o ) = 80o r = q = 80o vertical angles are equal Therefore: s = t = p = 50o

  36. Find the missing angle measures. Properties of Triangles p = q = r = 60o a = b= c = 60o d = 180 – 60 = 120o s = t = 180 - 43= 68.5o 2 e + 18 = a = 60 exterior angle = sum of remote interior angles e = 60 – 18 = 42o

  37. Find the missing angle measures. • Find the value of x • Find the value of y z • x is a base angle • 180 = x + x + 50 • 130 = 2x • x = 65° 2) y & z are remote interior angles and base angles of an isosceles triangle Therefore: y + z = x and y = z y + z = 80° y = z = 40°

  38. Find the missing angle measures. • Find the value of x • Find the value of y 1) ∆CDE is equilateral All angles = 60° Using Linear Pair <BCD = 70° x is the vertex angle x = 180 – 70 – 70 x = 40° 70° 60° 2) y is the vertex angle y = 180 – 100 y = 80°

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