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RAY­ a part of a line with one endpoint.
A
SYMBOL
B
AB
EXAMPLES
ANGLE
­ formed by two noncollinear rays (sides) that meet at a common endpoint (vertex).
3 ways to name an angle.
A
4
BAC or
CAB
ANGLE BISECTOR
­ a line, segment or a ray that divides an angle into two angles.
OPPOSITE RAYS ­ two rays with common endpoints that form a straight line.
B
A
C
Ray AB and Ray AC are opposite rays.
Title: Aug 19­2:18 PM (1 of 9)
EXAMPLES
Name the vertex and sides of each angle.
2)
1)
Vertex:T
Vertex: R
Sides: RS,
Sides: TS,
RQ
TU
Name each angle in three ways.
4)
3)
1
2
1
R
SRQ or
QRS
2
T
STU or
UTS
Name all angles that have B as a vertex.
5)
1
D
A
2
ABC or
C
1 2
CBA
B
H
LE bisects HLP
E
P
6) If m HLE = 37 ,
then m ELP = ____
37
L
LE bisects HLP
7) If m HLP = 84 ,
42
then m ELP = ____
H
E
P
L
LE bisects HLP
H
E
P
8) If m HLE = 39 ,
78
then m HLP = ____
L
Title: Aug 19­2:22 PM (2 of 9)
9) If m MPN = 2x + 14, and the m NPR = x + 34, find x and m NPR.
2x + 14 = x + 34
x + 14 = 34
x = 20
m
m
m
NPR = x + 34
NPR = 20 + 34
NPR = 54
QP and QR are opposite rays. QS bisects TQR.
T
S
R
Q
P
10) m RQS = 6x + 4, m SQT = 7x ­ 2, find m RQT.
Since QS bisects
TQR, that makes
SQT ≅
SQR
7x - 2 = 6x + 4
x-2=4
x=6
m
RQT = 2 (m RQS) or 2 (m SQT)
m
RQT = 2 (6x + 4)
m
RQT = 2 (6(6) + 4)
m
RQT = 2 (40)
m
RQT = 80
OR
m
RQT = 2 (7x - 2)
m
RQT = 2 (7(6) - 2)
m
RQT = 2 (42 - 2)
m
RQT = 2 (40)
m
RQT = 80
Title: Aug 19­2:30 PM (3 of 9)
QP and QR are opposite rays. QS bisects TQR.
T
S
R
Q
P
11) m PQT = 8x + 12, m TQR = 2x + 8, find m TQR.
Since QP and QR are opposiye rays, that makes
m
PQT + m TQR = 180
8x + 12 + 2x + 8 = 180
10x + 20 = 180
10x = 160
x = 16
Title: Aug 19­2:48 PM (4 of 9)
PQR a straight angle.
m
m
TQR = 2x + 8
TQR = 2(16) + 8
m
TQR = 32 + 8
m
TQR = 40
PRACTICE!
Name the vertex and sides of each angle.
1)
2)
Vertex: M
Vertex: D
Sides: DE
Sides: ML, MN
DC
Name each angle in three ways.
3)
3
D
CDE or
EDC
4
F
EFG or
GFE
4)
Name all angles that have V as a vertex.
5)
HVI
IVJ
HVJ
2
3
DVF
6)
7) Name the vertex of 2.
8) Name the sides of 4.
C
BA, BC
9) Write another name for BDC.
CDB or
Title: Aug 19­2:57 PM (5 of 9)
1
Classify each angle as right, acute, or obtuse.
10) WXY = obtuse angle
11) WXZ = acute angle
In the figure, QP and QR are opposite rays, and QT bisects RQS.
12) If m RQT = 6x + 5 and m SQT = 7x ­ 2,
find m RQT.
Since QT bisects
RQS, that makes
m
RQT = 6x + 5
m
RQT = 6(7) + 5
m
RQT = 42 + 5
m
RQT = 47
SQT ≅
RQT
7x - 2 = 6x + 5
x-2=5
x=7
In the figure, QP and QR are opposite rays, and QT bisects RQS.
13)Find m TQS if m RQS = 22a ­ 11 and m RQT = 12a ­ 8.
Since QT bisects
m
RQS = 2(m
RQS, that makes
RQT)
22a - 11 = 2(12a - 8)
22a - 11 = 24a - 16
- 11 = 2a - 16
5 = 2a
m
TQS = 12a - 8
m
TQS = 12(2.5) - 8
m
TQS = 30 - 8
m
TQS = 22
2.5 = a
Title: Aug 19­3:04 PM (6 of 9)
OR
m
SQT ≅
RQT
RQS = m
RQT
2
22a - 11 = 12a - 8
2
11a - 5.5 = 12a - 8
-5.5 = a - 8
2.5 = a
ANGLE PAIRS
ADJACENT ANGLES
­ are two angles that lie in the same plane, have a common vertex, and a common side, but no common
interior points.
VERTICAL ANGLES (measurement are always equal)
­ are two nonadjacent angles formed by two intersecting lines.
LINEAR PAIR (sum is always equal to 180)
­ is a pair of adjacent angles whose noncommon sides are opposite rays.
COMPLEMENTARY ANGLES
­ are two angles whose measures have a sum of 90 .
SUPPLEMENTARY ANGLES
­ are two angles whose measures have a sum of 180 .
PERPENDICULAR LINES
­ are lines that form right angles.
Title: Aug 19­3:32 PM (7 of 9)
PRACTICE!
ANGLE RELATIONSHIPS
ANSWER:
Linear Pair
ANSWER:
Adjacent
ANSWER:
Adjacent
ANSWER:
Adjacent
5)
ANSWER: Complementary
ANSWER:
ANSWER:
Vertical
Linear Pair
ANSWER:
Find the value of the variable.
9)
b = 180 - 50
b = 130
10)
b = 43
Title: Aug 19­3:35 PM (8 of 9)
Vertical
11)
b = 90 - 63
b = 27
14)
13)
12)
b = 180 - 35
b = 145
b = 90 - 29
b = 180 - 49
b = 61
b = 131
Find the value of the variable.
15)
17)
16)
3x + 18 + 93 = 180
OR
3x + 18 = 87
3x = 69
x = 23
6x + 2 + 40 = 90
2 + 3x = 62
3x = 60
x = 20
OR
6x + 2 = 50
6x = 48
x=8
18) If the ratio of the complementary angles is 2:1, what is the measure of the larger angle?
larger angle = 30 (2)
larger angle = 60
2x + x = 90
3x = 90
x = 30
19) If the ratio of the complementary angles is 7:2, what is the measure of the smaller angle?
smaller angle = 10 (2)
smaller angle = 20
21)
20)
m
7x + 2x = 90
9x = 90
x = 10
CBD + m
ABC = 90
m
LOM + m
MON = 90
4x - 3 + 5x + 39 = 90
4x + 52 + 8x - 10 = 90
9x + 36 = 90
12x + 42 = 90
9x = 54
12x = 48
x=6
x=4
m
CBD = 4x + 52
m
CBD = 4(4) + 52
m
CBD = 16 + 52
m
CBD = 68
m
LOM = 4x - 3
m
LOM = 4(6) - 3
m
LOM = 24 - 3
m
CBD = 21
Title: Aug 19­3:38 PM (9 of 9)