geometry unit 1: angles

GeometryUnit 1: Angles

ANGLES AND POINTSAn Angle is a figure formed by two rays with a common endpoint, called the vertex.

vertex

ray

rayAngles can have points in the interior, in the exterior or on the angle.

Example: Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex.

A

BC

DE

3 Ways to name an angleWe can name an angle using:

(1) Using 3 points

(2) Using 1 point

(3) Using a number

ABC or CBA∠ ∠

B∠

Using 3 points: vertex must be the middle letter

This angle can be named as

Using 1 point: using only vertex letter

* Use this method is permitted when the vertex point is the vertex of one and only one angle.

Since B is the vertex of only this angle, this can also be called .

A

B C

Naming an Angle

Naming an Angle - continued

2∠

Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named

as .

* The “1 letter” name is unacceptable when …more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.

2

A

B C

Example

K∠

, ,LKM PKM and LKP∠ ∠ ∠

32

K

L

M

P

Therefore, there is NO in this diagram.

There is2 3 5!!!There is also and but there is no∠ ∠ ∠

K is the vertex of more than one angle.

4 Types of Angles

Acute Angle: an angle whose measure is less than 90°.

Right Angle: an angle whose measure is exactly 90°.

Obtuse Angle: an angle whose measure is between 90° and 180°.

Straight Angle: an angle that is exactly 180°.

Measuring Angles Just as we can measure segments, we can also measure angles.

We use units called degrees to measure angles.

• A circle measures _____

• A (semi) half-circle measures _____

• A quarter-circle measures _____

• One degree is the angle measure of 1/360th of a circle.

360º

180º

90º

Adding Angles

When you want to add angles, use the notation m1, meaning the measure of ∠1.

If you add m∠1 + m∠2, what is your result?

m∠1 + m∠2 = 58°.

Therefore, m∠ADC = 58°.

m∠1 + m∠2 = m∠ADC also.

Angle Addition Postulate

R

M K

W

The sum of the two smaller angles will always equal the measure of the larger angle.

Complete:

m ∠ ____ + m ∠ ____ = m ∠ _____MRK KRW MRW

Postulate:

Example: Angle Addition

R

M K

W

3x + x + 6 = 90 4x + 6 = 90 – 6 = –64x = 84x = 21

K is interior to ∠MRW, m ∠ MRK = (3x), m∠ KRW = (x + 6) and m∠MRW = 90º. Find m∠MRK.

3xx+6 Are we done?

m∠MRK = 3x = 3•21 = 63º

First, draw it!

Angle Bisector

An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.

UK

53

Example: Since ∠4 ∠ 6, is an angle bisector.

Congruent Angles

∠3 ∠ 5.53

Definition: If two angles have the same measure, then they are congruent.

Congruent angles are marked with the same number of “arcs”.

The symbol for congruence is .

Example:

Example…Draw your own diagram and answer this question:

If is the angle bisector of ∠PMY and m∠PML = 87°, then find:

m∠PMY = _______

m∠LMY = _______

ML