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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

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

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

- 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

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

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

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

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

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

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

- When you want to add angles, use the notation m1, meaning the measure of 1.
- If you add m1 + m2, what is your result?
m1 + m2 = 58.

m1 + m2 = mADC also.

Therefore, mADC = 58.

Lesson 1-4: Angles

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 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

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

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

Example

- Draw your own diagram and answer this question:
- If is the angle bisector of PMY and mPML = 87, then find:
- mPMY = _______
- mLMY = _______

Lesson 1-4: Angles

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 m4 = 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: Ifm1 = 23 º and m2 = 32 º, find the measures of all other angles.

Answers:

Lesson 1-5: Pairs of Angles

Example: If m1 = 44º, m7 = 65º find the measures of all other angles.

Answers:

Lesson 1-5: Pairs of Angles

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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