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Warm Up Solve each equation. 1. 62 + x + 37 = 180 2. x + 90 + 11 = 180 3. 2 x + 18 = 180

Warm Up Solve each equation. 1. 62 + x + 37 = 180 2. x + 90 + 11 = 180 3. 2 x + 18 = 180 4. 180 = 3 x + 72. x = 81. x = 79. x = 81. x = 36. Common Core State Standards Learning Objective. 7.G.2 Construct triangles. Vocabulary. Triangle Sum Theorem acute triangle

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Warm Up Solve each equation. 1. 62 + x + 37 = 180 2. x + 90 + 11 = 180 3. 2 x + 18 = 180

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  1. Warm Up Solve each equation. 1. 62 + x + 37 = 180 2. x + 90 + 11 = 180 3. 2x + 18 = 180 4. 180 = 3x + 72 x = 81 x = 79 x = 81 x = 36

  2. Common Core State Standards Learning Objective 7.G.2 Construct triangles.

  3. Vocabulary Triangle Sum Theorem acute triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle midpoint altitude

  4. 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. You can also show this in a drawing.

  5. Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. Two 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°.

  6. An acute triangle has 3 acute angles. A right triangle has 1 right angle. An obtuse triangle has 1 obtuse angle.

  7. –117 –117 Additional Example 1: Finding Angles in Acute, Right and Obtuse Triangles A. Find p in the acute triangle. Triangle Sum Theorem 73° + 44° + p° = 180° 117 + p = 180 Subtract 117 from both sides. p = 63

  8. –85 –85 Additional Example 1: Finding Angles in Acute, Right, and Obtuse Triangles B. Find m in the obtuse triangle. 62 Triangle Sum Theorem 23° + 62° + m° = 180° m 23 85 + m = 180 Subtract 85 from both sides. m = 95

  9. –126 –126 Check It Out! Example 1 A. Find a in the acute triangle. Triangle Sum Theorem 88° + 38° + a° = 180° 38° 126 + a = 180 Subtract 126 from both sides. a = 54 88° a°

  10. 38° 24° c° –62 –62 Check It Out! Example 1 B. Find c in the obtuse triangle. Triangle Sum Theorem. 24° + 38° + c° = 180° 62 + c = 180 Subtract 62 from both sides. c = 118

  11. 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.

  12. –62 –62 2t = 118 2 2 Additional Example 2: Finding Angles in Equilateral, Isosceles, and Scalene Triangles A. Find the angle measures in the isosceles triangle. 62° + t° + t° = 180° Triangle Sum Theorem Simplify. 62 + 2t = 180 Subtract 62 from both sides. 2t = 118 Divide both sides by 2. t = 59 The angles labeled t° measure 59°.

  13. 10 10 Additional Example 2: Finding Angles in Equilateral, Isosceles, and Scalene Triangles B. Find the angle measures in the scalene triangle. 2x° + 3x° + 5x° = 180° Triangle Sum Theorem Simplify. 10x = 180 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°.

  14. –39 –39 2 2 Check It Out! Example 2 A. Find the angle measures in the isosceles triangle. 39° + t° + t° = 180° Triangle Sum Theorem Simplify. 39 + 2t = 180 Subtract 39 from both sides. 2t = 141 Divide both sides by 2 2t = 141 39° t = 70.5 t° t° The angles labeled t° measure 70.5°.

  15. Check It Out! Example 2 B. Find the angle measures in the scalene triangle. 3x° + 7x° + 10x° = 180° Triangle Sum Theorem 20x = 180 Simplify. 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 10(9°) = 90°. 10x° 3x° 7x°

  16. Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle measure. 12 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 figure.

  17. 10 10 Additional Example 3 Continued 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 figure. Triangle Sum Theorem x° + 6x° + 3x° = 180° Simplify. 10x = 180 Divide both sides by 10. x = 18

  18. Additional Example 3 Continued 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 figure. The angles measure 18°, 108°, and 54°. The triangle is an obtuse scalene triangle. x° = 18° 6 • 18° = 108° 3 • 18° = 54°

  19. Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle measures. 13 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.

  20. 5 5 Check It Out! Example 3 Continued 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. Triangle Sum Theorem x° + 3x° + x° = 180° Simplify. 5x = 180 Divide both sides by 5. x = 36

  21. 108° 36° 36° Check It Out! Example 3 Continued 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. The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle. x° = 36° 3 • 36° = 108° x° = 36°

  22. The midpoint of a segment is the point that divides the segment into two congruent segments. An altitude of a triangle is a perpendicular segment from a vertex of the triangle to the line containing the opposite side.

  23. Step 1 Find the length of TU. __ U 26 ft 1 2 1 2 T 20 ft S TU = UV = (20) = 10 V Additional Example 3: Finding the Length of a Line Segment __ __ In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST. __ __ T is the midpoint of UV.

  24. Step 2 Use the Pythagorean Theorem. Let ST = a and TU = b. __ The length of ST is 24 ft, or ST is 24 ft. Additional Example 3 Continued __ __ In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST. __ __ Pythagorean Theorem a2 + b2 = c2 Substitute 10 for b and 26 for c. a2 + 102 = 262 Simplify the powers. a2 + 100 = 676 –100 –100 Subtract 100 from each side. a2 = 576 a = 24 Find the square root.

  25. Step 1 Find the length of BC. __ C 25 in 1 2 1 2 B 14 in A = (14) = 7 BC = DC D Check It Out! Example 3 __ __ In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB. __ __ B is the midpoint of DC.

  26. Step 2 Use the Pythagorean Theorem. Let AB = a and BC = b. __ The length of AB is 24 in, or AB is 24 in. Additional Example 3 Continued __ __ In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB. __ __ Pythagorean Theorem a2 + b2 = c2 Substitute 7 for b and 25 for c. a2 + 72 = 252 Simplify the powers. a2 + 49 = 625 –49 –49 Subtract 49 from each side. a2 = 576 a = 24 Find the square root.

  27. 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°

  28. __ __ 5. In the figure, M is the midpoint of AB and MD is t perpendicular to AB. Find the length of AB. __ __ Lesson Quiz: Part II 3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°. 50° 4. Find the missing angle measure in an obtuse triangle with angle measures of 10° and 15°. 155° A 30 m 39 m 36 m D M B

  29. Classwork & Homework Lesson 8-4 Practice B Lesson 8-4 Problem Solving

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