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MODULE STUDY GUIDE REVIEW 19 Lines and Angles Essential Question: How can you use parallel and perpendicular lines to solve real-world problems? KEY EXAMPLE (Lesson 19.1) Find m∠ABD given that m∠CBE = 40° and the angles are formed ‹ › ‹ › − − by the intersection of the lines AC and DE at point B. When two lines intersect, they form two pairs of vertical angles at their intersection. Note that ∠ABD and ∠CBE are vertical angles and ∠DBC and ∠ABE are vertical angles. ∠ABD ≅ ∠CBE m∠ABD = m∠CBE = 40° Key Vocabulary vertical angles (ángulos verticales) complementary angles (ángulos complementarios) supplementary angles (ángulos suplementarios) transversal (transversal) indirect proof (prueba indirecta) Vertical Angles Theorem Definition of congruence of angles KEY EXAMPLE (Lesson 19.2) ‹ › ‹ › ‹ › − − − Find m∠APD given that AB intersects the parallel lines DE and FG at the points P and Q, respectively, and m∠AQF = 70°. When a transversal intersects two parallel lines, it forms a series of angle pairs. Note that ∠APD and ∠AQF are a pair of corresponding angles. m∠APD = m∠AQF m∠APD = 70° Corresponding Angles Theorem Substitute the known angle measure. © Houghton Mifflin Harcourt Publishing Company KEY EXAMPLE (Lesson 19.3) ‹ › ‹ › ‹ › − − − Determine whether the lines DE and FG are parallel given that AB intersects them at the points P and Q, respectively, m∠APE = 60°, and m∠BQF = 60°. ‹ › ‹ › − − Lines AB and DE intersect, so they create two pairs of vertical angles. The angle which is the opposite of ∠APE is ∠DPB, so they are called vertical angles. ∠APE ≅ ∠DPB m∠APE = m∠DPB m∠DPB = 60° Vertical Angles Theorem Definition of congruence Substitute the known angle measure. m∠BQF = m∠DPB = 60° ∠BQF ≅ ∠DPB Definition of congruence ‹ › ‹ › − − Thus, the lines DE and FG are parallel by the converse of the Corresponding Angles Theorem because their corresponding angles are congruent. Module 19 983 Study Guide Review EXERCISES Find the angle measure. 1. m∠ABD given that m∠CBD = 40° and the angles are formed by the intersection of the ‹ › ‹ › − − lines AC and DE at point B. (Lesson 19.1) ‹ › − 2. m∠BPE given that AB intersects the parallel ‹ › ‹ › − − lines DE and FG at the points P and Q, respectively, and m∠AQF = 45°. (Lesson 19.2) Determine whether the lines are parallel. (Lesson 19.3) ‹ › ‹ › ‹ › − − − 3. DE and FG , given that AB intersects them at the points P and Q, respectively, m∠APD = 60°, and m∠BQG = 120°. Find the distance and angle formed from the perpendicular bisector. (Lesson 19.4) 4. Find the distance of point D from _− _B given that D is the point at the perpendicular bisector of the ‹ › line segment AB, DE intersects AB, and AD = 3. Find m∠ADE. Find the equation of the line. (Lesson 19.5) 2 x + 2 and passes through the point (3, 4). 5. Perpendicular to y = __ 3 MODULE PERFORMANCE TASK Mystery Spot Geometry Inside mystery spot buildings, some odd things can appear to occur. Water can appear to flow uphill, and people can look as if they are standing at impossible angles. That is because there is no view of the outside, so the room appears to be normal. The illustration shows a mystery spot building constructed so that the floor is at a 25° angle with the ground. View from inside View from inside Use your own paper to complete the task. Use sketches, words, or geometry to explain how you reached your conclusions. Module 19 984 Study Guide Review © Houghton Mifflin Harcourt Publishing Company • A table is placed in the room with its legs perpendicular to the floor and the tabletop perpendicular to the legs. Sketch or describe the View from outside relationship of the tabletop to the floor, walls, and View from outside ceiling of the room. What would happen if a ball were placed on the table? • A chandelier hangs from the ceiling of the room. How does it appear to someone inside? How does it appear to someone standing outside of the room? Ready to Go On? 19.1–19.5 Lines and Angles • Online Homework • Hints and Help • Extra Practice ‹ › ‹ › − − Find the measure of each angle. Assume lines GB and FC are parallel. (Lessons 19.1, 19.2) 1. The measure of ∠WOX is 70°. Find m∠YOZ. B A X 2. The measure of ∠AXB is 40°. Find m∠FZE. H C W O Y 3. The measure of ∠XWO is 70°. Find m∠OYC. G 4. The measure of ∠BXO is 110°. Find m∠OZF. D Z F E PB is the perpendicular bisector of ¯ AC . ¯ QC is the Use the diagram to find lengths. ¯ ¯ perpendicular bisector of BD. AB = BC = CD. (Lesson 19.4) Q 5. Given BD = 24 and PC = 13, find PB. P © Houghton Mifflin Harcourt Publishing Company 6. Given QB = 23 and BC = 12, find QD. A Find the equation of each line. (Lessons 19.5) B C D 3 x + 5 and passing through the point (-7, -1) 7. The line parallel to y = -__ 7 1 x + 3 and passing through the point (2, 7) 8. The line perpendicular to y = __ 5 9. The perpendicular bisector to the line segment between (-3, 8) and (9, 4) ESSENTIAL QUESTION 10. Say you want to create a ladder. Which lines should be parallel or perpendicular to each other? Module 19 985 Study Guide Review MODULE 19 MIXED REVIEW Assessment Readiness 1. Consider each equation. Is it the equation of a line that is parallel or perpendicular to y = 3x + 2? Select Yes or No for A–C. 1x - 8 A. y = -_ Yes No 3 B. y = 3x - 10 Yes No C. y = 2x + 4 Yes No 2. Consider the following statements about △ABC. Choose True or False for each statement. C A. AC = BC True False B. CD = BC True False C. AD = BD True False A D B 3. The measure of angle 3 is 130° and the measure of angle 4 is 50°. State two different relationships that can be used to prove m∠1 = 130°. 4 1 3 2 4. m∠1 = 110° and m∠6 = 70°. Use angle relationships to show that lines m and n are parallel. 5 6 7 8 m n ℓ Module 19 986 Study Guide Review © Houghton Mifflin Harcourt Publishing Company 1 2 3 4