11 Chapter Introductory Geometry Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 11-4 More About Angles Constructing Parallel Lines The Sum of the Measures of the Angles of a Triangle The Sum of the Measures of the Interior Angles of a Convex Polygon with n sides The Sum of the Measures of the Exterior Angles of a Convex n-gon Walks Around Stars Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Vertical Angles Vertical angles created by intersecting lines are a pair of angles whose sides are two pairs of opposite rays. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles. Vertical angles are congruent. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Supplementary Angles The sum of the measures of two supplementary angles is 180°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Complementary Angles The sum of the measures of two complementary angles is 90°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Transversals and Angles Interior angles 2, 4, 5, 6 Exterior angles 1, 3, 7, 8 Alternate interior angles 2 and 5, 4 and 6 Alternate exterior angles 1 and 7, 3 and 8 Corresponding angles 1 and 2, 3 and 4, 5 and 7, 6 and 8 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Angles and Parallel Lines Property If any two distinct coplanar lines are cut by a transversal, then a pair of corresponding angles, alternate interior angles, or alternate exterior angles are congruent if, and only if, the lines are parallel. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Constructing Parallel Lines Place the side of triangle ABC on line m. Next, place a ruler on side AC. Keeping the ruler stationary, slide triangle ABC along the ruler’s edge until its side AB (marked A′B′ ) contains point P. Use the side to draw the line ℓ through P parallel to m. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
The Sum of the Measures of the Angles of a Triangle The sum of the measures of the interior angles of a triangle is 180°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-10 In the framework for a tire jack, ABCD is a parallelogram. If ADC of the parallelogram measures 50°, what are the measures of the other angles of the parallelogram? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-11 In the figure, m || n and k is a transversal. Explain why m1 + m 2 = 180°. Because m || n, angles 1 and 3 are corresponding angles, so m1 = m3. Angles 2 and 3 are supplementary angles, so m2 + m3 = 180°. Substituting m1 for m3, m1 + m2 = 180°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. The Sum of the Measures of the Interior Angles of a Convex Polygon with n sides The sum of the measures of the interior angles of any convex polygon with n sides is (n – 2)180°. The measure of a single interior angle of a regular n-gon is Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
The Sum of the Measures of the Exterior Angles of a Convex n-gon The sum of the measures of the exterior angles of a convex n-gon is 360°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-12 a. Find the measure of each interior angle of a regular decagon. The sum of the measures of the angles of a decagon is (10 − 2) · 180° = 1440°. The measure of each interior angle is Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-12 (continued) b. Find the number of sides of a regular polygon each of whose interior angles has measure 175°. Since each interior angle has measure 175°, each exterior angle has measure 180° − 175° = 5°. The sum of the exterior angles of a convex polygon is 360°, so there are exterior angles. Thus, there are 72 sides. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-13 Lines l and k are parallel, and the angles at A and B are as shown. Find x, the measure of BCA. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-13 (continued) Extend BC and obtain the transversal BC that intersects line k at D. The marked angles at B and D are alternate interior angles, so they are congruent and mD = 80°. mACD = 180° − (60° + 80°) = 40° x = mBCA = 180° − 40° = 140° Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Walks Around Stars The star can be obtained from a regular convex pentagon by finding its vertices as intersections of the lines containing the non-adjacent sides of the pentagon. The measure of each interior angle of the star is 36°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.