Geometry R/H1.4 – Angle Measures

1.5 – Angle Relationships

Line Segments

• A line segment is part of a line containing two endpoints and all points between them.

• Unlike lines, which extend forever in both directions, line segments have a definite beginning and end

• A line segment is named with the endpoints

A B

AB or BA

Rays

• A ray is part of a line that consists of one endpoint and all points of the line extending in one direction

• Name a ray using the endpoint first and then another point on the ray– When naming, make sure the arrow points away from the endpoint.

A

B

AB , not BA

Rays, continued

• Are these two rays the same?

A

B

A

B

No• Different endpoints• Extend in different directions

BA

AB

Opposite Rays

• Are two collinear rays with the same endpoint• Always form a line

JK

L

KJ and KL are opposite rays

Parallel Lines

• Lines that– Never intersect– Extend in the same directions– Coplanar– Have the same slope

Skew lines

Lines that:• Never intersect• Are noncoplanar• Extend in different directions

Parallel Planes

• Parallel planes are planes that never intersect• A line and plane that never intersect are also

parallel

Learning Check and Summary• Which of the following has no endpoints?

– Ray, Line Segment, Line• Which of the following has two endpoints?

– Ray, Line Segment, Line• Which of the following has one endpoint and

extends in one direction?– Ray, Line Segment, Line

• Which of the following extend in the same direction?– Parallel lines, skew lines

• Which of the following extend in different directions?– Parallel lines, skew lines

Types of Angles

Vertex

Side

Side

Naming Angles

• Angles are measured in degrees. The measure of is written as .

• Angles with the same measure are congruent.

1

Angle Addition

• If point B is the interior of , then

• If is a straight angle, then

AOCmBOCmAOBm

A

A

B

B

O

OC

C

PROTRACTOR EXERCISE(on Word doc)

What it is?

• Angle addition is adding (or subtracting) two (or more) Angles

• If ∠ABC is a right angle• m∠ABD is 50o

• find m∠DBC.

Variables (not scary!)

• If ∠ABC is a right angle• ∠ABD is (2x + 3)o

• m∠DBC is (x + 6)o

• solve for x

Example #1

• m∠1 = x Find x and find m 3• m∠2 = 2x – 10• m∠3 = 2x + 10

Solve for x

A

B

C

D

O1

2

3

4

Bisecting an Angle

• An angle bisector is a ray that divides an angle into two adjacent angles that are congruent.

• Ray FH Bisects Angle GFI because it divides the angle into two congruent angles.

• In the book, matching congruence arcs identify congruent angles in diagrams.

F

G

H

I

Which angles are adjacent? *Think: What does adjacent mean?

1 32

4

<1 & <2, <2 & <3, <3 & <4, <4 & <1

Vertical Angles – 2 angles that share a common vertex & whose sides form 2 pairs of opposite rays. Vertical Angles are congruent.

<1 & <3, <2 & <4

Then what do we call <1 & <3?

Linear Pair (of angles)• 2 adjacent angles whose non-common sides

are opposite rays.• The sum of a linear pair = 180 degrees

1 2

Ex 1: • Vertical angles?• Adjacent angles?• Linear pair? (2 adjacent angles=180 degrees)

• Adjacent angles not a linear pair?

1 3

2

5 4

Ex 2:

• If m<5=130o, findm<3m<6m<4

5 3

4

6

Ex 3:• Find the

values of x & y

&m<ABEm<ABDm<DBCm<EBC

3x+5o y+20ox+15o 4y-15o

A

B

C

D

E

Complementary Angles

• 2 angles whose sum is 90o

1

2

35o

A

55o

B<1 & <2 are complementary

<A & <B are complementary

Supplementary Angles

• 2 angles whose sum is 180o1

2

130o 50o

X Y

<1 & <2 are supplementary.

<X & <Y are supplementary.

Ex 4: <A & <B are supplementary. m<A is 5 times m<B. Find m<A & m<B.

m<A + m<B = 180o

m<A = 5(m<B)Now substitute!

5(m<B) + m<B = 180o

6(m<B)=180o

m<B=30o

m<A=150o