1. 11 Chapter Introductory Geometry
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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
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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.
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Supplementary Angles
The sum of the measures of two supplementary angles is 180°.
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Complementary Angles
The sum of the measures of two complementary angles is 90°.
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6. 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
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7. 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.
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8. 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.
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9. 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°.
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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?
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Example 11-11
In the figure, m || n and k is a transversal. Explain why m1 + m 2 = 180°.
Because m || n, angles 1 and 3 are corresponding angles, so m1 = m3.
Angles 2 and 3 are supplementary angles, so m2 + m3 = 180°.
Substituting m1 for m3, m1 + m2 = 180°.
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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
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13. 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°.
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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
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Example (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.
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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.
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Example (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 mD = 80°.
mACD = 180° − (60° + 80°) = 40°
x = mBCA = 180° − 40° = 140°
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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°.
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