1. Holt CA Course 1 8-4 Triangles Vocabulary Triangle Sum Theoremacute triangle right triangleobtuse triangle equilateral triangle isosceles triangle scalene trianglemidpoint altitude

2. Holt CA Course 1 8-4 Triangles A triangle is a three sided polygon. What are the different types of triangles? Equilateral Triangle: All Three Sides Are Congruent. Isosceles Triangle: Two Sides Are Of Equal Length. Scalene Triangle: No Sides Are Congruent Right Triangle: One Angle Is A Right Angle. Acute Triangle: All Angles Are Acute. Obtuse Triangle: One Angle is An Obtuse Angle. What is the formula for finding the area of a triangle? A = 1/2 bh b = base, h = height

3. Holt CA Course 1 8-4 Triangles

4. Holt CA Course 1 8-4 Triangles An acute triangle has 3 acute angles. A right triangle has 1 right angle. An obtuse triangle has 1 obtuse angle.

5. Holt CA Course 1 8-4 Triangles Additional Example 1: Finding Angles in Acute, Right and Obtuse Triangles A. Find p in the acute triangle. 73° + 44° + p° = 180° 117 + p = 180 p = 63 –117 Triangle Sum Theorem Subtract 117 from both sides.

6. Holt CA Course 1 8-4 Triangles Additional Example 1: Finding Angles in Acute, Right, and Obtuse Triangles B. Find m in the obtuse triangle. 23° + 62° + m° = 180° 85 + m = 180 m = 95 –85 Triangle Sum Theorem Subtract 85 from both sides. 23 62 m

7. Holt CA Course 1 8-4 Triangles Check It Out! Example 1 A. Find a in the acute triangle. 88° + 38° + a° = 180° 126 + a = 180 a = 54 –126 88° 38° a°a° Triangle Sum Theorem Subtract 126 from both sides.

8. Holt CA Course 1 8-4 Triangles B. Find c in the obtuse triangle. 24° + 38° + c° = 180° 62 + c = 180 c = 118 –62 –62 c°c° 24° 38° Check It Out! Example 1 Triangle Sum Theorem. Subtract 62 from both sides.

9. Holt CA Course 1 8-4 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.

10. Holt CA Course 1 8-4 Triangles Additional Example 2: Finding Angles in Equilateral, Isosceles, and Scalene Triangles 62° + t° + t° = 180° 62 + 2t = 180 2t = 118 –62 –62 A. Find the angle measures in the isosceles triangle. 2t = 118 2 t = 59 Triangle Sum Theorem Simplify. Subtract 62 from both sides. Divide both sides by 2. The angles labeled t° measure 59°.

11. Holt CA Course 1 8-4 Triangles Additional Example 2: Finding Angles in Equilateral, Isosceles, and Scalene Triangles 2x° + 3x° + 5x° = 180° 10x = 180 x = 18 10 10 B. Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 10. 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°.

12. Holt CA Course 1 8-4 Triangles Check It Out! Example 2 39° + t° + t° = 180° 39 + 2t = 180 2t = 141 –39 –39 A. Find the angle measures in the isosceles triangle. 2t = 141 2 t = 70.5 Triangle Sum Theorem Simplify. Subtract 39 from both sides. Divide both sides by 2 t° 39° The angles labeled t° measure 70.5°.

13. Holt CA Course 1 8-4 Triangles 3x° + 7x° + 10x° = 180° 20x = 180 x = 9 20 20 B. Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 20. 3x°3x°7x°7x° 10x° Check It Out! Example 2 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°.

14. Holt CA Course 1 8-4 Triangles 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. Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle measure. 1212 Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions

15. Holt CA Course 1 8-4 Triangles Additional Example 3 Continued x° + 6x° + 3x° = 180° 10x = 180 10 10 x = 18 Triangle Sum Theorem Simplify. Divide both sides by 10. 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.

16. Holt CA Course 1 8-4 Triangles x° = 18° 6 18° = 108° 3 18° = 54° The angles measure 18°, 108°, and 54°. The triangle is an obtuse scalene triangle. 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.

17. Holt CA Course 1 8-4 Triangles 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. Check It Out! Example 3 Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle measures. 1313

18. Holt CA Course 1 8-4 Triangles x° + 3x° + x° = 180° 5x = 180 5 5 x = 36 Triangle Sum Theorem Simplify. Divide both sides by 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.

19. Holt CA Course 1 8-4 Triangles x° = 36° 3 36° = 108° The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle. 36° 108° 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.

20. Holt CA Course 1 8-4 Triangles 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.

21. Holt CA Course 1 8-4 Triangles In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST. Additional Example 3: Finding the Length of a Line Segment 20 ft 26 ft S U V T Step 1 Find the length of TU. __ TU = UV 1212 T is the midpoint of UV. = (20) = 10 1212

22. Holt CA Course 1 8-4 Triangles In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST. Additional Example 3 Continued Step 2 Use the Pythagorean Theorem. Let ST = a and TU = b. __ Find the square root.a = 24 a 2 + b 2 = c 2 a 2 + 10 2 = 26 2 a 2 + 100 = 676 –100 –100 a 2 = 576 Pythagorean Theorem Substitute 10 for b and 26 for c. Simplify the powers. Subtract 100 from each side. The length of ST is 24 ft, or ST is 24 ft. __

23. Holt CA Course 1 8-4 Triangles In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB. Check It Out! Example 3 Step 1 Find the length of BC. __ BC = DC 1212 B is the midpoint of DC. = (14) = 7 1212 14 in 25 in A C D B

24. Holt CA Course 1 8-4 Triangles Additional Example 3 Continued Step 2 Use the Pythagorean Theorem. Let AB = a and BC = b. __ Find the square root.a = 24 a 2 + b 2 = c 2 a 2 + 7 2 = 25 2 a 2 + 49 = 625 –49 –49 a 2 = 576 Pythagorean Theorem Substitute 7 for b and 25 for c. Simplify the powers. Subtract 49 from each side. The length of AB is 24 in, or AB is 24 in. __ In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB.

25. Holt CA Course 1 8-4 Triangles Lesson Quiz: Part I 1. Find the missing angle measure in the acute triangle shown. 2. Find the missing angle measure in the right triangle shown. 38° 55°

26. Holt CA Course 1 8-4 Triangles Lesson Quiz: Part II 3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°. 4. Find the missing angle measure in an obtuse triangle with angle measures of 10° and 15°. 50° 155° 5. In the figure, M is the midpoint of AB and MD is t perpendicular to AB. Find the length of AB. __ 36 m 39 m D M A B 30 m