Holt CA Course Triangles Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview

Holt CA Course Triangles Warm Up Solve each equation x + 37 = x = x + 18 = = 3x + 72 x = 81 x = 79 x = 81 x = 36

Holt CA Course Triangles MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Also covered: Review of 6MG2.2 California Standards

Holt CA Course Triangles Vocabulary Triangle Sum Theoremacute triangle right triangleobtuse triangle equilateral triangle isosceles triangle scalene triangle

Holt CA Course 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.

Holt CA Course 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. You can also show this in a drawing.

Holt CA Course Triangles Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. The three angles in the triangle can be arranged to form a straight line or 180°. Two sides of the triangle are transversals to the parallel lines.

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

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

Holt CA Course 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

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

Holt CA Course 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.

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

Holt CA Course Triangles Additional Example 2: Finding Angles in Equilateral, Isosceles, and Scalene Triangles 2x° + 3x° + 5x° = 180° 10x = 180 x = 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°.

Holt CA Course Triangles Check It Out! Example 2 39° + t° + t° = 180° t = 180 2t = 141 –39 –39 A. Find the angle measures in the isosceles triangle. 2t = 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°.

Holt CA Course Triangles 3x° + 7x° + 10x° = 180° 20x = 180 x = 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°.

Holt CA Course 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 Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions

Holt CA Course Triangles Additional Example 3 Continued x° + 6x° + 3x° = 180° 10x = 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.

Holt CA Course 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.

Holt CA Course 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

Holt CA Course Triangles x° + 3x° + x° = 180° 5x = 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.

Holt CA Course 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.

Holt CA Course 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.

Holt CA Course 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) =

Holt CA Course 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 = 26 2 a = 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. __

Holt CA Course 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) = in 25 in A C D B

Holt CA Course 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 = 25 2 a = 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.

Holt CA Course 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°

Holt CA Course 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