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Angles. Lesson 1-4. Angle and Points. An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray. vertex. ray. Angles can have points in the interior, in the exterior or on the angle. A. E. D. B. C.

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Angles

Angles

Lesson 1-4

Lesson 1-4: Angles


Angle and points
Angle and Points

  • An Angle is a figure formed by two rays with a common endpoint, called the vertex.

ray

vertex

ray

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

A

E

D

B

C

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

B is the vertex.

Lesson 1-4: Angles


Naming an angle 1 using 3 points 2 using 1 point 3 using a number next slide
Naming an angle:(1) Using 3 points (2) Using 1 point (3) Using a number – next slide

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

C

B

Lesson 1-4: Angles


Naming an angle continued
Naming an Angle - continued

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 .

A

B

2

C

* 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.

Lesson 1-4: Angles


Example
Example

  • K is the vertex of more than one angle.

Therefore, there is NO in this diagram.

There is

Lesson 1-4: Angles


4 types of angles
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 .

Lesson 1-4: Angles


Complementary angles
Complementary Angles

Definition:

A pair of angles whose sum is 90˚

Examples:

Adjacent Angles

( a common side )

Non-Adjacent Angles

Lesson 1-5: Pairs of Angles


Supplementary angles
Supplementary Angles

Definition:

A pair of angles whose sum is 180˚

Examples:

Adjacent supplementary angles are also called “Linear Pair.”

Non-Adjacent Angles

Lesson 1-5: Pairs of Angles


Vertical angles
Vertical Angles

Definition:

A pair of angles whose sides form opposite rays.

Examples:

Vertical angles are non-adjacent angles formed by intersecting lines.

Lesson 1-5: Pairs of Angles


Theorem vertical angles are

Statements

Reasons

1.

1.

Definition: Linear Pair

2.

2.

Property: Substitution

3.

3.

Property: Subtraction

4.

4.

Definition: Congruence

Theorem: Vertical Angles are =

~

Given:

The diagram

Prove:

Lesson 1-5: Pairs of Angles


Adjacent angles
Adjacent Angles

Definition:

A pair of angles with a shared vertexand common sidebut do not have overlapping interiors.

Examples:

1 and 2 are adjacent. 3 and 4 are not.

1 and ADC are not adjacent.

4

3

Adjacent Angles( a common side )

Non-Adjacent Angles

Lesson 1-5: Pairs of Angles


Adding angles
Adding Angles

  • When you want to add angles, use the notation m1, meaning the measure of 1.

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

    m1 + m2 = 58.

m1 + m2 = mADC also.

Therefore, mADC = 58.

Lesson 1-4: Angles


Angle addition postulate
Angle Addition Postulate

Postulate:

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

Complete:

m  ____ + m ____ = m  _____

MRK

KRW

MRW

Lesson 1-4: Angles


Angle addition postulate1
Angle Addition Postulate

Postulate:

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

Complete:

m  ____ + m ____ = m  _____

MRK

KRW

MRW

Lesson 1-4: Angles


Angle bisector
Angle Bisector

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

Example:

Since 4   6, is an angle bisector.

5

3

Lesson 1-4: Angles


Congruent angles
Congruent Angles

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 

3

5

Example:

3   5.

Lesson 1-4: Angles


Example1
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 = _______

Lesson 1-4: Angles


What s important in geometry
What’s “Important” in Geometry?

4 things to always look for !

90˚

180˚

360˚

Most of the rules (theorems)

and vocabulary of Geometry

are based on these 4 things.

. . . andCongruence

Lesson 1-5: Pairs of Angles


Example if m 4 67 find the measures of all other angles
Example:If m4 = 67º, find the measures of all other angles.

Step 1: Mark the figure with given info.

Step 2: Write an equation.

67º

Lesson 1-5: Pairs of Angles


Example if m 1 23 and m 2 32 find the measures of all other angles
Example: Ifm1 = 23 º and m2 = 32 º, find the measures of all other angles.

Answers:

Lesson 1-5: Pairs of Angles


Example if m 1 44 m 7 65 find the measures of all other angles
Example: If m1 = 44º, m7 = 65º find the measures of all other angles.

Answers:

Lesson 1-5: Pairs of Angles


Algebra and geometry
Algebra and Geometry

Common Algebraic Equations used in Geometry:

( ) = ( )

( ) + ( ) = ( )

( ) + ( ) = 90˚

( ) + ( ) = 180˚

If the problem you’re working on has a variable (x),

then consider using one of these equations.

Lesson 1-5: Pairs of Angles


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