Name Class Date 1 C REFLECT. m 1 m 2 m 3 m 4 m 1 + m 2 m 2 + m 3 m 3 + m 4 m 4 + m 1
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1 Name lass Date 1-4 Pairs of ngles Going Deeper Essential question: How can you use angle pairs to solve problems? Recall that two rays with a common endpoint form an angle. The two rays form the sides of the angle, and the common endpoint marks the vertex. You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number. 1 ngle names:,,, 1 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. 1 G-O.1.1 E X P L O R E Measuring ngles Using a ruler, draw a pair of intersecting lines. Label each angle from 1 to 4. Sample answer: Use a protractor to help you complete the chart. 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 Sample answers given. Measure of ngle REFLET 1a. onjecture Share your results with other students. Make a conjecture about pairs of angles that are opposite of each other. Make a conjecture about pairs of angles that are next to each other. Sample answer: Opposite pairs of angles have the same measure. Pairs of angles that are next to each other have measures that add up to. hapter 1 21 Lesson 4
2 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 vertex 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. You have discovered in Explore 1 that adjacent angles formed by two intersecting lines are supplementary. 2 G-O.3.9 EXMPLE Identifying ngles and ngle Pairs Use the diagram below. D F E Sample answer: F and F Name a pair of adjacent angles. Name a pair of vertical angles. Name a pair of complementary angles. D Name an angle that is supplementary to FE. E Name an angle that is supplementary to FD. Sample answer: F F Name an angle that is supplementary to FD. F G Name a pair of non-adjacent angles that are complementary. Sample answer: F and DFE Sample answer: F and F F REFLET 2a. What is the measure of DFE? Explain how you found the measure. The measure of DFE is 40. ngles DFE and F are vertical angles; therefore, DFE also has a measure of 40. 2b. re F and DFE vertical angles? Why or why not? No, they are not vertical angles because they are not opposite angles formed by intersecting lines. 2c. re FD and FE vertical angles? Why or why not? Yes, they are vertical angles because they are opposite angles formed by intersecting lines. hapter 1 22 Lesson 4 DFE and F
3 3 -ED.1.1 E X M P L E Finding ngle Measures Find the measure of each angle. D 75 x D D and D are supplementary angles. The sum of their measures is. Write an equation to help you find the measure of D x = In the box, solve the equation for x. m D = x = x = 105 EHF F E 2x H 48 G EHF and FHG are supplementary angles. The sum of their measures is. In the box, write and solve an equation to help you find m EHF. REFLET m EHF = x = x = 132 3a. friend claims that two acute angles are complementary and their measures are 2x and (30-5x). If your friend is right, what equation must be true? 2x + (30-5x) = 90, because complementary angles are two angles whose measures have a sum of 90. 3b. Solve the equation you found and interpret the answer. Evaluate your friend s claim. 2x + (30-5x) = 90; 30-3x = 90, -3x = 60; x = -20. Substituting -20 for x into the expressions 2x and (30-5x) gives -40 and 130 as the angle measures. These are not two acute angle measures, so my friend s claim is false. hapter 1 23 Lesson 4
4 p r a c t i c e Use the figure for Exercises m QUP + m PUT = 2. Name a pair of supplementary angles. 90 T P Q Sample answer: SUR and QUR U 3. Name a pair of vertical angles. S Sample answer: SUR and PUQ R N 4. Name a pair of adjacent angles. Sample answer: SUR and SUP 5. What is the measure of QUN? Explain your answer. 90 ; sample answer: m SUT = 90, and m SUT and m NUQ are vertical angles. Vertical angles have the same measure. Solve for the indicated angle measure or variable. Y 6. m YL = 7. x = I S 22 L K 84 M 4x 8. 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? Yes, a parking lot can be built because the measure of angle J is 40, which is greater than 38. H G Green 50 J Park ve. ve. 9. Error nalysis student states that when the sum of two angle measures equals, the two angles are complementary. Explain why the student is incorrect. When two angle measures add up to the angles are supplementary, not complementary. hapter 1 24 Lesson 4
5 Name lass Date dditional Practice PQR and SQR form a linear pair. Find the sum of their measures. 2. Name the ray that PQR and SQR share. Use the figures for Exercises 3 and supplement of Z 4. complement of Y 5. n angle measures 12 degrees less than three times its supplement. Find the measure of the angle. 6. n angle is its own complement. Find the measure of a supplement to this angle. 7. DEF and FEG are complementary. m DEF = (3x 4), and m FEG = (5x + 6). Find the measures of both angles. 8. DEF and FEG are supplementary. m DEF = (9x + 1), and m FEG = (8x + 9). Find the measures of both angles. Use the figure for Exercises 9 and 10. In 2004, several nickels were minted to commemorate the Louisiana Purchase and Lewis and lark s expedition into the merican West. One nickel shows a pipe and a hatchet crossed to symbolize peace between the merican government and Native merican tribes. 9. Name a pair of vertical angles. 10. Name a linear pair of angles. 11. and D form a linear pair and have equal measures. Tell if is acute, right, or obtuse. 12. KLM and MLN are complementary. LM bisects KLN. Find the measures of KLM and MLN. hapter 1 25 Lesson 4
6 Problem Solving Use the drawing of part of the Eiffel Tower for Exercises Name a pair of angles that appear to be complementary. 2. Name a pair of supplementary angles. 3. If m SW = 45, what is m JST? How do you know? 4. If m FK = 135, what is m KL? How do you know? 5. Name three angles whose measures sum to. hoose the best answer. 6. landscaper uses paving stones for a walkway. Which are possible angle measures for a and b so that the stones do not have space between them? 50, , , 45 D 90, The angle formed by a tree branch and the part of the trunk above it is 68. What is the measure of the angle that is formed by the branch and the part of the trunk below it? F 22 H 158 G 112 J 8. R and S are complementary. If m R = (7 + 3x) and m S = (2x + 13), which is a true statement? R is acute. R and S are right angles. R is obtuse. D m S > m R hapter 1 26 Lesson 4
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