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Triangles Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Triangles Warm Up Solve each equation. 1. 62 + x + 37 = 180 x = 81 2. x + 90 + 11 = 180 x = 79 3. 2x + 18 = 180 x = 81 4. 180 = 2x + 72 + x x = 36 Triangles Problem of the Day What is the one hundred fiftieth day of a non-leap year? May 30 Triangles Learn to find unknown angles and identify possible side lengths in triangles. Triangles Vocabulary Triangle Sum Theorem acute triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Triangle Inequality Theorem Triangles If you tear off two corners of a triangle and place them next to the third corner, the three angles seem to form a straight line. Triangles Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. The sides of the triangle are transversals to the parallel lines. The three angles in the triangle can be arranged to form a straight line or 180°. Triangles An acute triangle has 3 acute angles. A right triangle has 1 right angle. An obtuse triangle has 1 obtuse angle. Triangles Additional Example 1A: Finding Angles in Acute, Right, and Obtuse Triangles Find c° in the right triangle. 42° + 90° + c° = 180° 132° + c° = 180° –132° –132° c° = 48° Triangles Additional Example 1B: Finding Angles in Acute, Right, and Obtuse Triangles Find m° in the obtuse triangle. 23° + 62° + m° = 180° 85° + m° = 180° –85° –85° m° = 95° Triangles Additional Example 1C: Finding Angles in Acute, Right and Obtuse Triangles Find p° in the acute triangle. 73° + 44° + p° = 180° 117° + p° = 180° –117° –117° p° = 63° Triangles Check It Out: Example 1A Find b in the right triangle. 38° 38° + 90° + b° = 180° 128° + b° = 180° –128° –128° b° = 52° b° Triangles Check It Out: Example 1B Find a° in the acute triangle. 88° + 38° + a° = 180° 38° 126° + a° = 180° –126° –126° a° = 54° a° 88° Triangles Check It Out: Example 1C Find c° in the obtuse triangle. 24° + 38° + c° = 180° 62° + c° = 180° –62° –62° c° = 118° 38° 24° c° Triangles An equilateral triangle has 3 congruent sides and 3 congruent angles. An isosceles triangle has at least 2 congruent sides and 2 congruent angles. A scalene triangle has no congruent sides and no congruent angles. Triangles Additional Example 2A: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find the angle measures in the isosceles triangle. 62° + t° + t° = 180° 62° + 2t° = 180° –62° –62° Triangle Sum Theorem Combine like terms. Subtract 62° from both sides. 2t° = 118° 2t° = 118° Divide both sides by 2. 2 2 t° = 59° The angles labeled t° measure 59°. Triangles Additional Example 2B: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find the angle measures in the scalene triangle. 2x° + 3x° + 5x° = 180° 10x° = 180° 10 10 Triangle Sum Theorem Combine like terms. Divide both sides by 10. x = 18° The angle labeled 2x° measures 2(18°) = 36°, the angle labeled 3x° measures 3(18°) = 54°, and the angle labeled 5x° measures 5(18°) = 90°. Triangles Additional Example 2C: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find the angle measures in the equilateral triangle. 3b° = 180° Triangle Sum Theorem 3b° 180° = 3 3 Divide both sides by 3. b° = 60° All three angles measure 60°. Triangles Check It Out: Example 2A Find the angle measures in the isosceles triangle. 39° + t° + t° = 180° Triangle Sum Theorem 39° + 2t° = 180° Combine like terms. –39° –39° Subtract 39° from both sides. 2t° = 141° 2t° = 141° Divide both sides by 2 2 2 39° t° = 70.5° The angles labeled t° measure 70.5°. t° t° Triangles Check It Out: Example 2B Find the angle measures in the scalene triangle. 3x° + 7x° + 10x° = 180° Triangle Sum Theorem 20x° = 180° Combine like terms. 20 20 Divide both sides by 20. x = 9° The angle labeled 3x° measures 3(9°) = 27°, the angle labeled 7x° measures 7(9°) = 63°, and the angle labeled 10x° measures 3x° 10(9°) = 90°. 10x° 7x° Triangles Check It Out: Example 2C Find the angle measures in the equilateral triangle. 3x° = 180° Triangle Sum Theorem 3x° 180° = 3 3 x° x° = 60° All three angles measure 60°. x° x° Triangles Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions The second angle in a triangle is six times as large as the first. The third angle is half as large as the second. Find the angle measures and draw a possible picture. Let x° = the first angle measure. Then 6x° = second angle measure, and 1 (6x°) = 3x° = 2 third angle measure. Triangles Additional Example 3 Continued Let x° = the first angle measure. Then 6x° = second angle measure, and 1 (6x°) = 3x° = 2 third angle. x° + 6x° + 3x° = 180° 10x° = 180° 10 10 x° = 18° Triangle Sum Theorem Combine like terms. Divide both sides by 10. Triangles Additional Example 3 Continued Let x° = the first angle measure. Then 6x° = second angle measure, and 1 (6x°) = 3x° = 2 third angle. x° = 18° 3 • 18° = 54° 6 • 18° = 108° The angles measure 18°, 54°, and 108°. The triangle is an obtuse scalene triangle. Triangles Check It Out: Example 3 The second angle in a triangle is three times larger than the first. The third angle is one third as large as the second. Find the angle measures and draw a possible figure. Let x° = the first angle measure. Then 3x° = second angle measure, and 1 (3x°) = x° = 3 third angle measures. Triangles Check It Out: Example 3 Continued Let x° = the first angle measure. Then 3x° = second angle measure, and 1 (3x°) = 3x° = 3 third angle. x° + 3x° + x° = 180° Triangle Sum Theorem 5x° = 180° 5 5 Combine like terms. Divide both sides by 5. x° = 36° Triangles Check It Out: Example 3 Continued Let x° = the first angle measure. Then 3x° = second angle measure, and 1 (3x°) = x° = 3 third angle. x° = 36° 3 • 36° = 108° x° = 36° The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle. 108° 36° 36° Triangles Additional Example 4A: Using the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 8 ft, 10 ft, 13 ft Find the sum of the lengths of each pair of sides and compare it to the third side. ? 8 + 10 > 13 18 > 13 ? 10 + 13 > 13 23 > 13 ? 8 + 13 > 10 21 > 10 A triangle can have these side lengths. The sum of the lengths of any two sides is greater than the length of the third side. Triangles Additional Example 4B: Using the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 2 m, 4 m, 6 m Find the sum of the lengths of each pair of sides and compare it to the third side. ? 2+4>6 6>6 A triangle cannot have these side lengths. The sum of the lengths of two sides is not greater than the length of the third side. Triangles Check It Out: Example 4 Tell whether a triangle can have sides with the given lengths. Explain. 17 m, 15 m, 33 m Find the sum of the lengths of each pair of sides and compare it to the third side. ? 17 + 15 > 33 32 > 33 A triangle cannot have these side lengths. The sum of the lengths of two sides is not greater than the length of the third side. Triangles Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems Triangles Lesson Quiz: Part I 1. Find the missing angle measure in the acute triangle shown. 38° 2. Find the missing angle measure in the right triangle shown. 55° Triangles Lesson Quiz: Part II 3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°. 50° 4. Tell whether a triangle can have sides with lengths of 4 cm, 8 cm, and 12 cm. No; 4 + 8 is not greater than 12 Triangles Lesson Quiz for Student Response Systems 1. Identify the missing angle measure in the acute triangle shown. A. 43° B. 57° C. 80° D. 90° Triangles Lesson Quiz for Student Response Systems 2. Identify the missing angle measure in the acute triangle shown. A. 40° B. 50° C. 90° D. 180° Triangles Lesson Quiz for Student Response Systems 3. Identify the missing angle measure in an acute triangle with angle measures of 38° and 61°. A. 38° B. 61° C. 81° D. 99°

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