1. Lesson 1-4
Angles
Lesson 1-4: Angles

2. Angle and Points ray vertex ray
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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

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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

4. Naming an Angle - continued
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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

5. Example Therefore, there is NO in this diagram.
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Example
K is the vertex of more than one angle.
Therefore, there is NO in this diagram.
There is
Lesson 1-4: Angles

6. 4 Types of Angles Acute Angle: Right Angle: Obtuse Angle:
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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

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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
Lesson 1-4: Angles

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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

9. Angle Addition Postulate
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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

10. Example: Angle Addition
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Example: Angle Addition
K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK.
First, draw it!
3x + x + 6 = 90
4x + 6 = 90
– 6 = –6
4x = 84
x = 21 3x
x+6
Are we done?
mMRK = 3x = 3•21 = 63º
Lesson 1-4: Angles

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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

12. Congruent Angles Definition:
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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

13. Example Draw your own diagram and answer this question:
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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