Warm Up

1 ft.

Find the area of the green region.

Assume all angles are right angles.

2 ft.

2 ft.

2 ft.

1 ft.

1 ft.1 ft.

2 ft.

4 ft.

3 ft.

1 ft.

1 ft.

Sections 3.6 and 3.7

Congruent Angles

• Definition: they have the same degree measure.

• Symbols: angle A congruent angle B if and only if m a and m b

• Picture: 30°• B

30°

A

The first relationship we are going to talk about

Definition: Two angles are vertical angles if their sides form two pairs of opposite rays

Angles 1 and 2 are vertical angles 1

2

3 4Angles 3 and 4 are also vertical angles

Vertical angles are always congruent.

Theorem 3.1-Vertical Angle Thm

• Definition: vertical angles are congruent

• Picture:

• Symbols: Angles 1 and 3 are congruent

• Angles 2 and 4 are congruent

2

41 3

5y – 50

4y – 10

What type of angles

are these?

5y – 50 = 4y – 10 y = 40

Plug y back into our angle equations and we get

150

What is the measure of the angle?

Find the value of x in each figure

• 1. 2.

• 3. 4.

130°

x°

5x° 25°

x° 40°(x – 10)°

125 °

Theorem 3-2:

• If two angles are congruent, then their complements (90 °) are congruent.

• Picture:• 60° 60°

• The measure of angles complementary to A and B is 30.

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example 1:

1 and 2

ADJACENT

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example 2:

VERTICAL

1 and 4

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example 3:

ADJACENT,

COMPLEMENTARY

3 and 4

Theorem 3-3:

• If two angles are congruent, then their supplements (180 °) are congruent.

• 70° 110° 110° 70°

• Angle 1 is congruent to angle 4 (measure of 70°)

Theorem 3-4:

• If two angles are complementary to the same angle, then they are congruent.

• Angle 3 is complementary to angle 4, angle 5 is complementary to angle 4. Angle 3 is congruent to angle 5.

34 5

Theorem 3.5:

• If two angles are supplementary to the same angle, then they are congruent.

• Angle 1 is supplementary to angle 2. Angle 3 is supplementary to angle 2. Angle 1 congruent to angle 3.

13

2

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example 4:

ADJACENT,

SUPPLEMENTARY,

LINEAR PAIR

1 and 5

Find x, y, and z.

Example 5:

51xy

z

x = 129,

y = 51,

z = 129

Find x.

Example 6:

X = 8

( (5 3x x - 15) = + 1) 5 15 3 1x x 2 15 1x 2 16x

(3x + 1)

L

P AT

O

(5x - 15) (20x - 5)

Find

Example 7:

155

m LAT(3x + 1)

L

P AT

O

(5x - 15) (20x - 5)

Since we have already found the value of x, all we need to do now is to

plug it in for LAT.

20 5 20 8 5x ( )160 5

Theorem 3-6:

• If two angles are congruent and supplementary, then each is a right angle.

• 1 2

Theorem 3-7:

• All right angles are congruent.

• These angles are congruent

Moving right along…

• Section 3-7- Perpendicular Lines

• This is the last section of the chapter! YEAH.

Lines that intersect to form four right

angles are perpendicular lines.

m

l1 2

4 3Symbol:

m l is read as m perpendicular to l

Clipper- Flying Cloud Ship

• The main mast and the frame for the sails are examples of perpendicular line segments. The main mast is perpendicular to the sail frame, the likewise, the frame for the sail is perpendicular to the main mast.

Flying Cloud

• One common nineteenth century ship was the clipper. This ship, which had many sails, was designed for speed. In fact, it was named a clipper because of the way it “clipped off” the miles.

Theorem 3-8:

• If two angles are perpendicular, then they form four right angles.

Each angle is 90 degrees.

Each angle is congruent.

12

3

456

B

G V

F

A

C

E

8

GV AE&'&''&''&''&''&''&''&''&''&''&''&''&''&' '&'

12

3

456

B

G V

F

A

C

E

8

AE FV&'&''&''&''&''&''&''&''&''&''&''&''&''&' '&'

12

3

456

B

G V

F

A

C

E

8

4 1

12

3

456

B

G V

F

A

C

E

8

3 4

12

3

456

B

G V

F

A

C

E

8

1 2 90

12

3

456

B

G V

F

A

C

E

8

is a

right

GVA

12

3

456

B

G V

F

A

C

E

8

12

3

456

B

G V

F

A

C

E

8

6 & 3 are

supplementary

12

3

456

B

G V

F

A

C

E

8

6 & 2 are

complementary

Theorem 3.9:

• If a line m is in a plane and point T is a point on m, then there exists exactly one line in that plane that is perpendicular to m at T. m

°

°T

Homework: workbook page 17 and 18 ALL