Chapter Introductory Geometry 11 Copyright © 2013, 2010, and 2007, Pearson Education, Inc

Chapter

Introductory Geometry

1111

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


Vertical Angles

Angles 1 and 3 are vertical angles.

Angles 2 and 4 are vertical angles.

Vertical angles are congruent.

Vertical angles created by intersecting lines are a pair of angles whose sides are two pairs of opposite rays.


Supplementary Angles

The sum of the measures of two supplementary angles is 180°.


Complementary Angles

The sum of the measures of two complementary angles is 90°.


Transversals and Angles

Interior angles 2, 4, 5, 6Exterior angles 1, 3, 7, 8Alternate interior angles 2 and 5, 4 and 6Alternate exterior angles 1 and 7, 3 and 8

Corresponding angles 1 and 2, 3 and 4, 5 and 7, 6 and 8


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.


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.


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°.


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?


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.

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

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°.


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


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.


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.


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.

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°.


Documents

Chapter Introductory Geometry 11 Copyright © 2013, 2010, and 2007, Pearson Education, Inc