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Line and Angle Relationships Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Line and Angle Relationships Warm Up Find the complement of each angle measure. 1. 30° 60° 2. 42° 48° Find the supplement of each angle measure. 4. 82° 98° 3. 150° 30° Line and Angle Relationships Problem of the Day Draw three points that are not on the same line. Label them A, B, and C. How many lines can you draw that are determined by the points? Name the lines. 3; AB, AC, BC Line and Angle Relationships Learn to identify parallel, perpendicular, and skew lines, and angles formed by a transversal. Line and Angle Relationships Vocabulary perpendicular lines parallel lines skew lines adjacent angles vertical angles transversal Line and Angle Relationships When lines, segments, or rays intersect, they form angles. If the angles formed by two intersecting lines measure 90°, the lines are perpendicular lines. Some lines in the same plane do not intersect at all. These lines are parallel lines. Segments and rays that are part of parallel lines are also parallel. Skew lines do not intersect, and yet they are also not parallel. They lie in different planes. Line and Angle Relationships Reading Math The symbol means “is parallel to.” The symbol means “is perpendicular to.” Line and Angle Relationships Additional Example 1A: Identifying Parallel, Perpendicular, and Skew Lines Tell whether the lines appear parallel, perpendicular, or skew. UV and YV UV YV The lines appear to intersect to form right angles. Line and Angle Relationships Additional Example 1B: Identifying Parallel, Perpendicular, and Skew Lines Tell whether the lines appear parallel, perpendicular, or skew. XU and WZ XU and WZ are skew. The lines are in different planes and do not intersect. Line and Angle Relationships Additional Example 1C: Identifying Parallel, Perpendicular, and Skew Lines Tell whether the lines appear parallel, perpendicular, or skew. XY and WZ XY || WZ The lines are in the same plane and do not intersect. Line and Angle Relationships Check It Out: Example 1A Tell whether the lines appear parallel, perpendicular, or skew. WX and XU WX XU The lines appear to intersect to form right angles. Line and Angle Relationships Check It Out: Example 1B Tell whether the lines appear parallel, perpendicular, or skew. WX and UV WX and UV are skew. The lines are in different planes and do not intersect. Line and Angle Relationships Check It Out: Example 1C Tell whether the lines appear parallel, perpendicular, or skew. WX and ZY WX || ZY The lines are in the same plane and do not intersect. Line and Angle Relationships Adjacent angles have a common vertex and a common side, but no common interior points. Angles 2 and 3 in the diagram are adjacent. Adjacent angles formed by two intersecting lines are supplementary Vertical angles are the opposite angles formed by two intersecting lines. Angles 1 and 3 in the diagram are vertical angles. Vertical angles have the same measure, so they are congruent. Line and Angle Relationships Reading Math Angles with the same number of tick marks are congruent. The tick marks are placed in the arcs drawn inside the angles. Line and Angle Relationships A transversal is a line that intersects two or more lines. Transversals to parallel lines form special angle pairs. Line and Angle Relationships Line and Angle Relationships Additional Example 2A: Using Angle Relationships to Find Angle Measures Line n line p. Find the measure of the angle. 2 2 and the 130° angle are vertical angles. Since vertical angles are congruent, m2 = 130°. Line and Angle Relationships Additional Example 2B: Using Angle Relationships to Find Angle Measures Line n line p. Find the measure of the angle. 3 m3 + 130° = 180° –130° –130° m3 = 50° Adjacent angles formed by two intersecting lines are supplementary. Subtract 130° to isolate m3. Line and Angle Relationships Additional Example 2C: Using Angle Relationships to Find Angle Measures Line n line p. Find the measure of the angle. 4 Alternate interior angles are congruent. m4 = 130°. Line and Angle Relationships Check It Out: Example 2A Line n line p. Find the measure of the angle. 45° 3 4 5 6 2 3 135° 7 n p 3 and the 45° angle are vertical angles. Since vertical angles are congruent, m3 = 45°. Line and Angle Relationships Check It Out: Example 2B Line n line p. Find the measure of the angle. 45° 6 4 5 6 2 3 135° 7 n p 6 and the 135° angle are vertical angles. m6 = 135°. Line and Angle Relationships Check It Out: Example 2C Line n line p. Find the measure of the angle. 45° 4 5 6 2 3 135° 7 4 n m4 + 45° = 180° –45° –45° m4 = 135° p Adjacent angles formed by two intersecting lines are supplementary. Subtract 45° to isolate m4. Line and Angle Relationships Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems Line and Angle Relationships Lesson Quiz Tell whether the lines appear parallel, perpendicular, or skew. 1. AB and CD parallel 2. EF and FH perpendicular 3. AB and CG skew 4. In Exercise 28, line r || line s. Find the measures of 4, 5, and 7. 55°, 125°, 125° Line and Angle Relationships Lesson Quiz for Student Response Systems 1. Use the figure to identify the type of the given lines. A. parallel B. perpendicular C. skew D. none Line and Angle Relationships Lesson Quiz for Student Response Systems 2. Use the figure to identify the type of the given lines. A. parallel B. perpendicular C. skew D. none Line and Angle Relationships Lesson Quiz for Student Response Systems 3. Use the figure to identify the type of the given lines. A. parallel B. perpendicular C. skew D. none Line and Angle Relationships Lesson Quiz for Student Response Systems 4. In the figure, line x || line y. Identify the measures of 2, 6, and 7. A. 70°, 110°, 70° B. 110°, 70°, 70° C. 110°, 110°, 70° D. 70°, 70°, 110°