Name Date Class. Vertical angles are opposite angles formed by the intersection of two lines. Vertical angles are congruent.

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1 SKILL 43 Angle Relationships Example 1 Adjacent angles are pairs of angles that share a common vertex and a common side. Vertical angles are opposite angles formed by the intersection of two lines. Vertical angles are congruent. Vocabulary Adjacent angles Vertical angles Congruent Complementary angles Supplementary angles Adjacent angles: 1 and 2, 2 and 3, 3 and 4, 4 and 1 Vertical angles: 1 and 3, 2 and 4 Also, 1 3 and 2 4. Classify each pair of figures as adjacent or vertical angles and and and and and and 4 7. If m 4 = 80, find m If m 3 = 100, find m

2 Example 2 The sum of the measures of complementary angles is 90. The sum of the measures of supplementary angles is 180. m 5 m 6 = 90 m 7 m 8 = 180 Use the figure. Describe each pair of angles as complementary, supplementary, or neither and and and 5 Find the measure of x in each figure Explain the Error. Find the correct solution. Each angle has a measure of 90, so the angles are complementary. 505

3 Example 3 The measure of 3 is 23. Find the measure of 1. 5 and 3 are vertical angles, so m 5 = 23 m 3 = 23 1 and 5 are complementary angles, so m 1 m 5 = 90. m 1 m 5 = 90 m 1 23 = 90 m 5 = 23 m 1 = m 1 = 67 Subtract 23 from each side. Simplify. The measure of 6 is 72. Find the measure of each angle. Explain your answer. 15. m m m m m 5 506

4 20. P and Q are adjacent angles. P and R are vertical angles. The measure of Q is 36 and the measure of R is 121. What is the measure of P? 21. Write an equation to represent the sum of the measures of ACB, BCD, and ECD. Then find m BCD. 22. A and B are complementary angles. The measure of A is (3x) and the measure of B is 45. What is the value of x? 23. Two vertical angles are supplementary. What is the measure of each angle? 24. Angles S and T are complementary angles. If m S is x and m T is (x 10), find the measure of each angle. SKILL 43 ANGLE RELATIONSHIPS 507

5 SKILL 49 Parallel Lines Cut by a Transversal Example 1 A transversal is a line that intersects two or more lines in a plane. In the figure, line t is a transversal that intersects line m and line n. Certain types of angle pairs are formed when a transversal intersects two lines. Same-side interior angles lie on the same side of the transversal and inside the intersected lines. Examples: 3 and 6, 4 and 5 Same-side exterior angles lie on the same side of the transversal and outside the intersected lines. Examples: 1 and 8, 2 and 7 Vocabulary Transversal Same-side interior angles Same-side exterior angles Alternate interior angles Alternate exterior angles Corresponding angles Parallel lines Alternate interior angles lie on opposite sides of the transversal and inside the intersected lines. Examples: 4 and 6, 3 and 5 Alternate exterior angles lie on opposite sides of the transversal and outside the intersected lines. Examples: 1 and 7, 2 and 8 Corresponding angles lie on the same side of the transversal and are in the same position in relation to the two intersected lines. Examples: 1 and 5, 2 and 6, 3 and 7, 4 and 8 Use the figure. Classify each pair of angles as same-side interior, same-side exterior, alternate interior, alternate exterior, or corresponding angles and and and and and 8 556

6 Example 2 Parallel lines lie in the same plane and never intersect. In the figure, line p is parallel to line q. Line p is parallel to line q is written as p q. When parallel lines are cut by a transversal, the following angle pairs are congruent. Alternate interior angles: 4 6, 3 5 Alternate exterior angles: 1 7, 2 8 The arrows on the line are used to indicate parallel li Corresponding angles: 1 5, 2 6, 3 7, 4 8 In the figure above, m 1 = 110. What other angles have a measure of 110? Since 1 and 5 are corresponding angles, m 5 = 110. Since 1 and 7 are alternate exterior angles, m 7 = 110. Since 7 and 3 are corresponding angles, m 3 = 110. In the figure, m 2 = 70. What other angles have a measure of 70? Since 2 and 6 are corresponding angles, m 6 = 70. Since 2 and 8 are alternate exterior angles, m 8 = 70. Since 8 and 4 are corresponding angles, m 4 = 70. In the figure, a b, m 1 = 82 and m 2 = 98. Find each angle measure. Explain your reasoning. 6. m 3 7. m 4 8. m 5 9. m Explain the Error. Find the correct solution. In the figure, line t is a transversal and m 4 = 65. Since 4 and 8 are corresponding angles, 4 8, so m 8 =

7 Example 3 When parallel lines are cut by a transversal, the angles formed are either congruent or supplementary. If you know the measure of one of the angles, you can find the measures of all of the angles. In the figure, m 1 = and 2 are supplementary. So, m 2 = 84 1 and 3 are vertical angles. So, m 3 = and 4 are supplementary. So, m 4 = 84. You can find the measures of 5, 6, 7, and 8 by identifying corresponding, alternate interior, alternate exterior, or supplementary angles. In the figure, f g, m 1 = 112 and m 16 = 78. Find each angle measure. 11. m m m m m m m m m m m m m m

8 25. Find the values for x and y. Explain your reasoning. Use the following information for The figure below shows the rungs of a ladder. In the figure, AB CD EF. 26. Is ACD AEF? Explain your reasoning. 27. Is ACD ABD? Explain your reasoning. 28. In the figure m n and r t. The lines intersect to form a parallelogram. What are the angle measures of the interior angles of the parallelogram? SKILL 49 PARALLEL LINES CUT BY A TRANSVERSAL 559

9 SKILL 50 Parallelograms Example 1 A parallelogram is a special type of quadrilateral. One property of parallelograms is that its opposite sides are congruent and parallel. In parallelogram ABCD, AB CD, BC AD, AB CD and AD BC. Vocabulary Parallelogram Quadrilateral Consecutive angles Diagonal Bisect Find the values for x and y. AB and CD are opposite sides, so AB CD. AB = CD Write an equation. 6 = x AB = 6, CD = x AD and BC are opposite sides, so AB CD. AD = BC Write an equation. 2y = 5 AD = 2y, BC = 5 y = 2.5 Divide each side by 2. So, x = 6 and y = 2.5. Find the value of x for each parallelogram Find the perimeter of parallelogram JKLM, if KL = 8 meters and LM = 6 meters. 566

10 4. Explain the Error. Find the correct solution. In parallelogram DEFG, DE FG and EF DG. If DE = 11 feet and DG = 14 feet, then EF = 11 feet and FG = 14 feet. 5. PQ and RS are opposite sides of a parallelogram. If PQ = 84 inches and RS = 6x, what is the value of x? Example 2 Another property of parallelograms is that their opposite angles are congruent. In parallelogram ABCD, A C and B D. If m A = 105, find m C. A and C are opposite angles, so A C. m A = m C Write an equation. 105 = m C m A = 105 If m B = (5x 35) and m D = 75, find the value of x. B and D are opposite angles, so B D. m B = m D Write an equation. 5x 35 = 75 m B = 5x 35 and m D = 75 5x = 40 Subtract 35 from each side. x = 8 Divide each side by 5. Use x = 8 to check that m B = 75. m B = 5x 35 m B = 5(8) 35 m B = 75 So, m A = m C = 105 and m B = m D =

11 For 6 9, use parallelogram ABCD. Find each measure. 6. m C 7. m D 8. AB 9. DA Find the value of x in each parallelogram Find the values of x and y in parallelogram DEFG. For 13 and 14, use parallelogram PQRS. 13. If m P = (2x 15), m Q = 117, and m R = 63, find the value for x. 14. If PQ = (5y 6) feet, RS = 19 feet, and SP = 24 feet, find the value for y. 568

12 Example 3 Consecutive angles of a parallelogram are supplementary. Two interior angles of a parallelogram are consecutive if they have a common side. In the parallelogram ABCD, the following pairs of angles are supplementary. A and B, B and C, C and D, D and A So, m A m B = 180, m B m C = 180, m C m D = 180, and m D m A = 180. If m A = 98, find the measures of the remaining angles. A and C are opposite angles, so A C. m A = m C = 98 A and B are consecutive angles. The sum of consecutive angles is 180. m A m B = 180 Write an equation. 98 m B = 180 m A = 82 m B = 82 Subtract 82 from each side. B and D are opposite angles, so B D. m B = m D = 82 So, m A = 98, m B = 82, m C = 98 and m D = 82. Use the given angle to name two pairs of consecutive angles. 15. S 16. T 17. U 18. V 19. Two angles of a parallelogram measure 57. What is the measure of the other two angles? 569

13 Find the unknown angle measures in each parallelogram Find the value of x. Then find the measure of each angle in the parallelogram. Example 4 Diagonals of a parallelogram bisect each other. In the parallelogram ABCD below, E is the midpoint of AC and BD. So, AE EC and BE ED. Suppose AE = 5x 14 and EC = 3x 10. Solve for x. AE = EC Write an equation. 5x 14 = 3x 10 AE = 5x 14 and EC = 3x 10 2x 14 = 10 2x = 24 Subtract 3x from each side. Add 14 to each side. x = 12 Divide each side by 2. Find the length of AE and EC. Replace x with 12 in the equations AE = 5x 14 and EC = 3x 10. AE = 5(12) 14 or 46 EC = 3(12) 10 or

14 For 23 and 24, use parallelogram GHIJ. 23. Find the values for x and y. 24. What is the length of each diagonal? For 25 27, use the following information. In parallelogram CDEF, FG = 2x 4, FD = 20, CG = 3y 2, and GE = 5y Find the values for x and y. 26. What is the length of GD? 27. What is the length of CE? SKILL 50 PARALLELOGRAMS 571

15 SKILL 58 Writing Equations of Parallel, Perpendicular, Vertical and Horizontal Lines Example 1 Lines that do not intersect are parallel lines. Parallel lines always have the same slope. A line is given by the equation y = 3x 4. Write a line parallel to this line going through point (1, 2). Vocabulary Horizontal Parallel Perpendicular Vertical Step 1: Step 2: Step 3: Identify the needed slope. Substitute the slope and coordinates of the point into y = mx b. Then solve for b. Now write the equation of the parallel line. m = 3 y = mx b 2 = 3(1) b 2 = 3 b 2 3 = 3 3 b 5 = b y = 3x 5 Write the equation of the line that is parallel to the given line and goes through the given point. 1. Line given by y = 4x 5 through the point ( 3, 0) 2. Line given by y = 2x 4 through the point (1, 3) 3. Line given by y = 1 x 2 through the point (8, 7) 2 636

16 Example 2 Lines that intersect at right angles are called perpendicular lines. Perpendicular lines have slopes that are negative reciprocals. A line is given by the equation y = 2x 5. Write a line perpendicular to this line going through point (2, 4). Step 1: Identify the needed slope. 1 m = 2 Step 2: Step 3: Substitute the slope and coordinates of the point into y = mx b. Then solve for b. Now write the equation of the parallel line. y = mx b 1 4 = 2 b 2 4 = 1 b 3 = b 1 y = x 3 2 Write the equation of the line that is perpendicular to the given line and goes through the given point. 4. Line given by y = 1 x 4 through the point ( 2, 9) 2 5. Line given by y = x 5 through the point (4, 2) 6. Line given by y = 3x 9 through the point (6, 4) 637

17 Example 3 Different vertical lines are always parallel to each other. Different horizontal lines are always parallel to each other. A vertical line and a horizontal line are always perpendicular to each other. Name a line parallel to y = 3 that passes through the point (4, 2). Choose the y-coordinate of the ordered pair to create the parallel line. y = 2 Name a line parallel to x = 2 that passes through the point ( 3, 1). Choose the x-coordinate of the ordered pair to create the parallel line. x = 2 Name a line perpendicular to the line y = 3 that passes through the point ( 2, 1). The line given by y = 3 is horizontal. Choose the x-coordinate to create the perpendicular line. x = 2 Write the equation of the line that is parallel to the given line and goes through the given point. 7. Line given by y = 6 through the point ( 3, 0) 8. Line given by x = 1 through the point (4, 5) Write the equation of the line that is perpendicular to the given line and goes through the given point. 9. Line given by y = 0 through the point (7, 8) 10. Line given by x = 3 through the point (4, 1) 638

18 11. Find the Error. Explain the solution. Question: Write the equation of the line 1 that is perpendicular to y = x 2 and 3 goes through (6, 1). Answer: The slope is 1 3 so the perpendicular slope is 3. Substituting gives 1 = 3(6) b or b = 17. The equation is y = 3x Write an equation that is parallel to y = 5x 4 and goes through (10, 0). 13. Write an equation that is perpendicular to y = 5x 4 and goes through (1, 0). 14. A company rents movies that can be streamed online or downloaded to a computer. The company charges a one-time sign-up fee and a charge for each movie x. The equation to represent this is y = 5x 20. What is the sign-up fee and cost per movie? 15. The owner is advertising a special in which the cost per movie stays the same but the sign-up fee decreases. For example, a customer will pay a total of $13 to rent two movies. A Write an equation that is parallel to y = 5x 20 and goes through point (2, 13). B What is the new sign-up fee? SKILL 58 WRITING EQUATIONS OF PARALLEL, PERPENDICULAR, VERTICAL, AND HORIZONTAL LINES 639