9.1 Angle Relationships
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1 ? LESSON 9.1 ngle Relationships ESSENTIL QUESTION How can you use angle relationships to solve problems? Equations, epressions, and relationships Write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships. EXPLORE TIVITY Measuring ngles It is useful to work with pairs of angles and to understand how pairs of angles relate to each other. ongruent angles are angles that have the same measure. STEP 1 STEP 2 Using a ruler, draw a pair of intersecting lines. Label each angle from 1 to 4. Use a protractor to help you complete the chart. Houghton Mifflin Harcourt Publishing ompany ngle m 1 m 2 m 3 m 4 m 1 + m 2 m 2 + m 3 m 3 + m 4 m 4 + m 1 Measure of ngle Reflect 1. onjecture Share your results with other students. Make a conjecture about pairs of angles that are opposite each other. 2. onjecture When two lines intersect to form two angles, what conjecture can you make about the pairs of angles that are net to each other? Lesson
2 Math On the Spot ngle Pairs and One-Step Equations Vertical angles are the opposite angles formed by two intersecting lines. Vertical angles are congruent because the angles have the same measure. djacent angles are pairs of angles that share a verte and one side but do not overlap. omplementary angles are two angles whose measures have a sum of 90. Supplementary angles are two angles whose measures have a sum of 180. You discovered in the Eplore ctivity that adjacent angles formed by two intersecting lines are supplementary. EXMPLE Use the diagram. Math Talk Mathematical Processes re D and E vertical angles? Why or why not? Name a pair of vertical angles. and DE Name a pair of adjacent angles. and D Name a pair of supplementary angles. and D 50 E D D ind the measure of. My Notes Use the fact that and D in the diagram are supplementary angles to find m. Reflect m + m D = = = 40 The measure of is 40. They are supplementary angles. m D = = 140 Subtract 140 from both sides. 3. nalyze Relationships What is the relationship between and? Eplain. Houghton Mifflin Harcourt Publishing ompany 4. Draw onclusions re and adjacent angles? Why or why not? 284 Unit 5
3 YOUR TURN Use the diagram. 5. Name a pair of supplementary angles. Personal Math Trainer Online ssessment and Intervention 6. Name a pair of vertical angles. E 35 G 7. ind the measure of GD. D ngle Pairs and Two-Step Equations Sometimes solving an equation is only the first step in using an angle relationship to solve a problem. EXMPLE Math On the Spot ind the measure of EH. EH and HG form a straight line E H G Houghton Mifflin Harcourt Publishing ompany STEP 1 STEP 2 Identify the relationship between EH and HG. EH and HG are supplementary angles. Write and solve an equation to find. m EH + m HG = = = 132 = 66 Since angles EH and HG form a straight line, the sum of the measures of the angles is 180. The sum of the measures of supplementary angles is 180. Subtract 48 from both sides. Divide both sides by 2. Since m EH = 2, then m EH = 132. Lesson
4 ind the measure of ZXY. STEP 1 STEP 2 Identify the relationship between WXZ and ZXY. WXZ and ZXY are complementary angles. W X 35 Z (4 + 7) Write and solve an equation to find. m WXZ + m ZXY = (4 + 7) = = The sum of the measures of complementary angles is 90. Substitute the values. ombine like terms. Subtract 42 from both sides. 4 = 48 = 12 Divide both sides by 4. Y STEP 3 ind the measure of ZXY. m ZXY = (4 + 7) = (4(12) + 7) = 55 Substitute 12 for. Use the Order of Operations. The measure of ZXY is 55. YOUR TURN 8. ind the measure of JML. L = m JML = 3 = 3 54 J M N 9. ritique Reasoning ory says that to find m JML above you can stop when you get to the solution step 3 = 126. Eplain why this works. Houghton Mifflin Harcourt Publishing ompany Personal Math Trainer Online ssessment and Intervention 286 Unit 5
5 Using ngle Measures in Triangles You learned earlier that the sum of the measures of the angles in any triangle is 180. You can use this property in many real-world situations. EXMPLE 3 The front of the top story of a house is shaped like an isosceles triangle. The measure of the angle at the top of the triangle is 70. ind the measure of each of the base angles Math On the Spot STEP 1 Make a sketch. 70 STEP 2 Write an equation. m + m + m = = 180 The sum of the angle measures in a triangle is 180. Substitute values. Houghton Mifflin Harcourt Publishing ompany Image redits: hristian eier/pictures/lamy STEP 3 Solve the equation to find = = = YOUR TURN Use the diagram. = = 55 Each of the base angles measures ind the value of. ombine like terms. Subtract 70 from both sides. Divide both sides of the equation by ind the measures of and. 3 Personal Math Trainer Online ssessment and Intervention Lesson
6 Guided Practice or Eercises 1 2, use the figure. (Eample 1) 1. Vocabulary The sum of the measures of UWV and UWZ is 90, so UWV and UWZ are angles. V W X Y 2. Vocabulary UWV and VWX share a verte and one side. They do not overlap, so UWV and VWX are U Z angles. or Eercises 3 4, use the figure. 3. G and DGE are angles, so m DGE =. (Eample 1) 4. ind the measure of EG. (Eample 2) m GD + m DGE + m EG = G 50 2 D E + + = = = m EG = 2 = 5. ind the measures of and. (Eample 3) m + m + m = = ? 2 + = = =, so m = =, so m =. ESSENTIL QUESTION HEK-IN 6. Suppose that you know that T and S are supplementary, and that m T = 3 (m S). How can you find m T? 40 Houghton Mifflin Harcourt Publishing ompany 288 Unit 5
7 Name lass Date 9.1 Independent Practice or Eercises 7 11, use the figure. T P Personal Math Trainer Online ssessment and Intervention Solve for each indicated angle measure or variable in the figure. S 41 R U N Q K I 84 M 4 H G 7. Name a pair of adjacent angles. Eplain why they are adjacent m KMH Solve for each indicated angle measure or variable in the figure. 8. Name a pair of acute vertical angles. 9. Name a pair of supplementary angles. 62 D 10. Justify Reasoning ind m QUR. Justify your answer. 14. m E E Houghton Mifflin Harcourt Publishing ompany 11. Draw onclusions Which is greater, m TUR or m RUQ? Eplain. 15. m 16. m Solve for each indicated angle measure or variable in the figure. P Q R m Q Lesson
8 OUS ON HIGHER ORDER THINKING Work rea Let be a right triangle with m = ritical Thinking n equilateral triangle has three congruent sides and three congruent angles. an be an equilateral triangle? Eplain your reasoning. 20. ountereample n isosceles triangle has two congruent sides, and the angles opposite those sides are congruent. River says that right triangle cannot be an isosceles triangle. Give a countereample to show that his statement is incorrect. 21. Make a onjecture In a scalene triangle, no two sides have the same length, and no two angles have the same measure. Do you think a right triangle can be a scalene triangle? Eplain your reasoning. 22. Represent Real-World Problems The railroad tracks meet the road as shown. The town will allow a parking lot at angle J if the measure of angle J is greater than 38. an a parking lot be built at angle J? Why or why not? Green 50 J Park ve. 23. nalyze Relationships In triangle XYZ, m X = 30, and all the angles have measures that are whole numbers. ngle Y is an obtuse angle. What is the greatest possible measure that angle Z can have? Eplain your answer. ve. Houghton Mifflin Harcourt Publishing ompany 290 Unit 5
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