9-3 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

Walks Around Stars and Other Figures

Vertical Angles – are pairs of angles such as 1ے and 3ے and appear any time two lines intersect.

Angles 1 and 3 are vertical angles. m 1ے = m 3ے

Angles 2 and 4 are vertical angles. m 2ے = m 4ے

Vertical angles are congruent.

m( 1ے ) = 180° - m( 2ے )

m( 3ے ) = 180 - (° m 2ے )

Supplementary Angles

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

Example #2 Page 609Find the measure of the third angle in each of the following triangles.

70° 50°

180° – (70° + 50°) = 60°

The measures of the interior angles of every triangle add to 180°. Subtract the sum of the given angles from 180° to find the third angle.

Example #2 Page 609Find the measure of the third angle in each of the following triangles.

60°

180° – (60° + 60°) = 60°

The measures of the interior angles of every triangle add to 180°. Subtract the sum of the given angles from 180° to find the third angle.

60°

Example #2 Page 609Find the measure of the third angle in each of the following triangles.

30°

180° – (90° + 30°) = 60°

The measures of the interior angles of every triangle add to 180°. Subtract the sum of the given angles from 180° to find the third angle.

Example #2 Page 609Find the measure of the third angle in each of the following triangles.

45°

180° – (90° + 45°) = 45°

The measures of the interior angles of every triangle add to 180°. Subtract the sum of the given angles from 180° to find the third angle.

Complementary Angles

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

Example # 4 Page 610Two angles are complementary and the ratio of their measures is 7:2 What are the angle measure?

The given ratio could be written 7x : 2xThe angles must add to 90°

7x + 2x = 90°9x = 90°x = 10

The angles are:

7(10) = 70°

and

2(10) = 20°

The Sum of the Measures of the Angles of a Triangle

Theorem 9-4The sum of the measures of the interior angles of a triangle is 180°.

Example # 6 Page 610In the following figure, DE || BC, EF || AB, and DF || AC, also m( 1ے ) = 45° andm( 2ے ) = 65°. Find each of the following values.

a) m( 3ے ) = 180° - [m( 1ے ) + m( 2ے )]= 180° - [45° + 65°]= 70°

b) m(ےD)ABCD is a parallelogram, thus opposite angles are equal m(ےD) = m( 3ے ) = 70°

D

A

E

C

F

B

1

23

45°

65°

Example # 6 Page 610In the following figure, DE || BC, EF || AB, and DF || AC, also m( 1ے ) = 45° andm( 2ے ) = 65°. Find each of the following values.

c) m(ےE)AECB is a parallelogram, thus m(ےE) = m( 2ے ) = 65°

d) m(ےF)ACFB is a parallelogram, thus m(ےF) = m( 1ے ) = 45 °

D

A

E

C

F

B

1

23

45°

65°

Transversals is any line that intersects a pair of lines in a plane.

Interior angles 2, 4, 5, 6Exterior angles 1, 3, 7, 8Alternate 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)

Theorem 9-3

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 line are parallel.

Constructing Parallel Lines

Example # 8 Page 610For each of the following, determine x if the lines a and b are parallel.

20°

30°

x

b

a

Answer:

Extend the 30° ray from line a to line b

A triangle is formed. The interior angles are 20°, 30°(alternate interior angles) and 130° (total 180° interior angles), x and 130° are supplementary, x = 50°

Example # 8 Page 610For each of the following, determine x if the lines a and b are parallel.

135°

140°

x

b

a

Answer:

Extend the 135° ray from line b to line a

The triangle is formed. The interior angles are 45° (supplement of 135°), 40° (supplement of 140°) and 95° (total of 180° interior angles) x and 95° are supplements, x = 85°

Example # 8 Page 610For each of the following, determine x if the lines a and b are parallel.

70°

30°

x

b

a

Answer:

The interior angles of the triangle are 30°, 110° (supplement of 70° - corresponding angles) and x = 40° (total of 180° interior angles)

Example # 8 Page 610For each of the following, determine x if the lines a and b are parallel.

25°

120°

x

b

a

Answer:

Extend the 25° ray from line b to line a

A quadrilateral is formed , with a total of (n – 2)(180°) = 360° interior angles. The interior angles are 120°, 90°, 25° (alternate interior angles) and 125° (total 360° interior angles) x is supplementary to 125°, so x = 55°

Example # 10a Page 610In each of the following, a relationship between the marked angles is given. In each case prove k || l

150°

150°

l

k

Answer:

The marked angles are alternate exterior angles. k || l by theorem 9-3.

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 line are parallel.

Example # 10b Page 610In each of the following, a relationship between the marked angles is given. In each case prove k || l

y

z

l

k

Answer:

Extend the yx line and label as w.

x = 180° - [180° - (z + w)]

x = z + w

It is given that x = y + z , so z + w = y + z, therefore w = y

Because w and y are measures of alternate interior angles k || l

x

x = y + zw

Example # 10c Page 610In each of the following, a relationship between the marked angles is given. In each case prove k || l

y l

k

Answer:

Label the angles supplementary to y as x1 Then x1 + y = 180°, and since x + y = 180° it follows that x = x1 Since x and x1 are measures corresponding angles, therefore k || l

x

x + y = 180°

x1

Example # 10d Page 610In each of the following, a relationship between the marked angles is given. In each case prove k || l

w l

k

From part (b), z = w + v and y = u + x, thus u = y – x and v = z – w It is given that y = x + z – w, so y – x = z – w, Therefore, u = y – x = z – w = v, Since u and v are measures of alternate interior, k || l

y

y = x + z - wx

z

Answer:

Extend the yz line and label as v.

v

u

The sum of the measures of the exterior angles of a convex polygon is 360°.

Theorem 9-5

Example # 12 a Page 610Find the sum of the measures of the marked angles in each of the following figures.

1

2

3 4

5

6

The six angles surrounding the center point sum to 360° They can be compared to three pairs of vertical angles with the angles contained by triangles equal to those not contained.

There are two of each angle making up the circle around the intersection point (the included angle and its vertical angle)

Since the included angle and its vertical angle are always congruent, the contained angles must sum to half of 360° = 180°

The sum of the angles in the three triangles is 3 * 180° = 540°

Thus the numbered angles sum to 540° - 180° = 360°

Example # 12 b Page 610Find the sum of the measures of the marked angles in each of the following figures.

1

2

3

4

5

6

m( 1ے ) = m( 3ے ) = m( 5ے ) = 60° (an equilateral triangle)

Thus m( 1ے ) + m( 3ے ) + m( 5ے ) = 180°

Likewise m( 2ے ) + m( 4ے ) + m( 6ے ) = 180°

Example # 12 c Page 610Find the sum of the measures of the marked angles in each of the following figures.

1

2

3

4

56

The sum of the measures of the interior angles in each large triangle is 180°

180° + 180° = 360°

The Sum of the Measures of the Interior Angles of a Convex Polygon with n sides

The sum of the measure of the exterior angle is 360°, so the sum of the measures of the interior angles is n . 180° - 360°, or (n – 2)180°.

The measure of a single interior angle of a regular n-gon is or ( 2)180n

n

-

n

n 360180

Examplea) Find the measure of each interior angle of a regular decagon.

Because a decagon has 10 sides, all of which are congruent, the sum of the measures of the angles is (10 x 180) – 360 = 1440° so each interior angle has a measure of 1440° ÷ 10 = 144° and each exterior angle is 360° ÷ 10 = 36°

b) Find the number of sides of a regular polygon each of whose interior angles has measure 135°.

Each interior angle of the regular polygon is 135° The measure of each exterior angle of the polygon is 180° - 135° = 45° Because the sum of the measures of all exterior angles if a convex polygon is 360° the number of exterior angles is 360/45 = 8 TTherefore the number of sides is 8.

Solution:

Extend BC to D. Then BCA is an exterior angle of ADC. So mx = 180 – 140 = 40°.

The sum of the angles in a triangle is 180°

so my = 180° – (60° + 40°) = 80°.

EBD BDA and they are corresponding angles, so line l ┴ line k.

Example: Prove that lines l and k are parallel.

Walks Around Stars and Other Figures

The sum of the measures of the exterior angles is 5x = 360k.k is some positive integer

Example # 14 Page 611Two sides of a regular octagon are extended as shown in the following figure. Find the measure of 1ے

1

The measure of each interior angle of a regular octagon =

00(8 2) 180

1358

The triangle has two equal interior angles of 180°-135° = 45° (supplementary angles)

Thus m( 1ے ) = 180) -°2 * 45 = (°90 °

HOMEWORK 9-3

Page 609 - 612

1, 3, 5, 7, 9, 11, 13, 15