Download Lesson 2: Angles Angles are formed when two points branch out

Lesson 2: Angles
Angles are formed when two points branch out from a single point. An angle is the
union of two rays. The angles formed by vertical and horizontal lines are called right
angles; lines, segments, or rays that intersect in right angles are said to be
perpendicular.
Angles can be measured in either degrees (0) (from 0 to 360) or radians (from 0
to 2π). Angles length can be determined by measuring along the arc they map
out on a circle. In radians we consider the length of the arc of the circle
mapped out by the angle. Since the circumference of a circle is 2π, a right
angle is
900.
radians. In degrees, the circle is 3600, and so a right angle would be
For the purposes of this course, most of the time we will talk about
measurement in degrees (0)
Angles are named in several ways.
(1) By naming the vertex of the angle
A
B
(2) By naming a point on each side of the angle with the vertex in between.
B
(3) By placing a small number on the interior of the angle near the vertex.
a
Classification of angles by degree measure:
Acute Angle: An angle is said to be acute if it measures between 00 and 900,
exclusive.
(acute angle)
C
Right : An angle is said to be right if it measures 900.
Notice the small box placed in the corner of a right angle, unless the box is present it is
assumed the angle is not 900.
*all right angles are congruent
Notice the box placed in the angle!
(right angle)
Obtuse Angle: An angle is said to be obtuse if it measures between 900 and 180 0,
exclusive.
(obtuse angle)
Straight: An angle is said to be straight if it measures exactly 1800.
(straight angle)
Reflex: An angle is said to be reflex if it measures between 1800 and 3600 exclusively.
B
C
A
(reflex angle)
Special angle pairs
Adjacent angles are angles with a common (shared) vertex and a common (shared) side.
Adjacent angles have no interior points in common.
Complementary angles are two angles whose sum is 90 0.
Complementary angles may or may not be adjacent. If two complementary angles are
adjacent, then their exterior sides are perpendicular (forming a right angle).
A
C
650
B
250
650 + 250 = 900
ABC and
CBD are complementary
ABC and
CBD are adjacent
D
Supplementary angles are two angles whose sum is 180
0
Supplementary angles may or may not be adjacent.
Linear pairs are formed when two angles are both adjacent and supplementary.
ABC is adjacent to
A
B
C
D
CBD
ABD and
CBD are supplementary
ABD and
CBD are linear pairs.
Vertical angles sharing a common vertex whose sides form opposite rays are called vertical
angles. Vertical angles are congruent (equal).
ray
ray
A
B
C
D
A=
B
A and
B are vertical angles
C=
D
C and
D are vertical angles
B and
C
are adjacent angles
A and
D are adjacent angles
Lesson 2 Review
1. Identify each of the following angle measures as acute, right, obtuse, straight, or reflex.
a)
b)
c)
d)
e)
f)
g)
35°
164°
193°
90°
45°
68°
359°
91°
1810
240
1790
930
570
930
650
2050
380
1010
810
490
310
280
610
1800
320
940
210
190
2. Identify the following as adjacent, complementary, supplementary, linear pairs, or vertical
angles. Some angles may have more than one characteristic. Justify, or prove, your
answers. In other words, if you state that two angles are supplementary, then the proof is
because their sums equal 1800.
a.
b.
c.
A
B
102
d.
D
78°
e.
f.
A
b
C
B
A
Cd
B
B
C
D
A
H
D
E
G
F
3. Why do complementary or supplementary angles not have to be adjacent? Why are
vertical angles always congruent? Justify your answers.
4. Find the measures of the following complementary angles.
Angle A
Angle B.
64°
?
36°
?
42°
?
51°
?
68°
?
5. Find the measures of the following supplementary angles.
If Angle A measures
The Angle B measures
126°
?
132°
?
178°
?
45°
?
89°
?