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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. - PowerPoint PPT Presentation
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Lesson 1-4: Angles 1
Lesson 1-4
Angles
Lesson 1-4: Angles 2
Angle and Points
An Angle is a figure formed by two rays with a common endpoint, called the vertex.
vertex
ray
ray Angles can have points in the interior, in the exterior or on the
angle.
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
Lesson 1-4: Angles 3
Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide
ABC or CBA
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
B
Lesson 1-4: Angles 4
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 .2
* 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
Lesson 1-4: Angles 5
Example
K
32
K
L
M
P
Therefore, there is NO in this diagram.There is , ,LKM PKM and LKP
2 3 5!!!There is also and but there is no
K is the vertex of more than one angle.
Lesson 1-4: Angles 6
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 7
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 8
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.
22°
36°
21
D
B
C
A
Therefore, mADC = 58.
m1 + m2 = mADC also.
Lesson 1-4: Angles 9
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:
Lesson 1-4: Angles 10
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 mMRW = 90º. Find mMRK.
3xx+6 Are we done?
mMRK = 3x = 3•21 = 63º
First, draw it!
Lesson 1-4: Angles 11
Angle Bisector
An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.
UK
j41°
41°
64
U
K53
Example: Since 4 6, is an angle bisector.
Lesson 1-4: Angles 12
3 5.
Congruent Angles
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:
Lesson 1-4: Angles 13
Example
Draw your own diagram and answer this question: If is the angle bisector of PMY and mPML = 87,
then find: mPMY = _______ mLMY = _______
ML
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