Transcript Chapter 1 Section 1

B
Refer to the figure below.
1
G
2
3
A
D
C
1) Which angles have D as their vertex?
2) Give two other names for <ABD.
3) Does <BDC appear to be acute, right, obtuse, or straight?
4) Find the value of x and m<ABD if m<ABC = 71, m<DBC = 2x –
3, and m<ABD = 3x + 4.
5) Suppose MN bisects <RMQ, m<RMN = x + 2, and m<NMQ = 2x –
3. Find the value of x and m<RMQ.
B
Refer to the figure below.
1
G
2
3
A
D
C
1) Which angles have D as their vertex?
<ADB and <BDC
2) Give two other names for <ABD.
<DBA and <1
3) Does <BDC appear to be acute, right, obtuse, or straight?
Obtuse
B
Refer to the figure below.
1
G
2
3
A
D
C
4) Find the value of x and m<ABD if m<ABC = 71, m<DBC = 2x –
3, and m<ABD = 3x + 4.
Use the angle addition postulate.
m<ABD + m<DBC = m<ABC
3x + 4 + 2x – 3 = 71
5x + 1 = 71
5x = 70
x = 14
Plug 14 in for x in the equation for m<ABD.
m<ABD = 3x + 4
m<ABD = 3(14) + 4
m<ABD = 42 + 4
m<ABD = 46
5) Suppose MN bisects <RMQ, m<RMN = x + 2, and m<NMQ = 2x
– 3. Find the value of x and m<RMQ.
R
N
M
Q
Since MN bisects <RMQ, we know that m<RMN = m<NMQ.
m<RMN = m<NMQ
x + 2 = 2x – 3
2=x–3
5=x
Plug x in to either equation and multiple by two(angle
addition postulate).
m<RMQ = 2(m<RMN)
m<RMQ = 2(x + 2)
m<RMQ = 2x + 4
m<RMQ = 2(5) + 4
m<RMQ = 10 + 4
m<RMQ = 14
Angle Relationships
Perpendicular lines- Lines that
intersect to form right angles.
Adjacent Angles- Angles in the same
plane that have a common vertex
and a common side, but no common
interior points. (<1 and <2 are
adjacent angles)
Vertical Angles- Two nonadjacent
angles formed by two intersecting
lines. (<3 and <4 are vertical angles)
Vertical angles are congruent.
1
2
3
4
Linear Pair- Adjacent angles
whose noncommon sides are
opposite rays.( <1 and <2 form a
linear pair)
1
2
Supplementary angles- Two
angles whose measures have a
sum of 180. (<3 and <4 are
supplementary)
Complementary angles- Two
angles whose measures have a
sum of 90. (<5 and <6 are
complementary)
3
5
6
4
Example 1: Use the figure below to find the value of x and
J
m<JIH.
G
16x - 20
K
I
13x + 7
H
Since <GIJ and <KIH are vertical angles, we know they are
congruent.
m<GIJ = m<KIH
16x – 20 = 13x + 7
3x – 20 = 7
3x = 27
x=9
Since <KIH and <JIH are supplementary, we know they add up to
180. Plug 9 in for x in the equation for <KIH.
m<KIH + m<JIH = 180
13x + 7 + m<JIH = 180
13(9) + 7 + m<JIH = 180
117 + 7 + m<JIH = 180
124 + m<JIH = 180
m<JIH = 56
Example 2: Suppose the measure of angle GIJ = 9x – 4 and
the measure of angle JIH = 4x – 11. Find the value of x and
the measure of angle KIH.
J
G
9x - 4
4x - 11
I
K
H
Since <GIJ and <JIH are supplementary and form a linear pair,
their measures add up to 180.
m<GIJ + m<JIH = 180
9x – 4 + 4x – 11 = 180
13x – 15 = 180
13x = 195
x = 15
Since <KIH and <GIJ are vertical angles, they are congruent or
have equal measures. Plug 15 in for x in the equation for m<GIJ.
m<GIJ = 9x – 4
m<GIJ = 9(15) - 4
m<GIJ = 135 – 4
m<GIJ = 131 = m<KIH
Example 3: The measure of the complement of an angle is 3.5
times smaller than the measure of the supplement of the angle.
Find the measure of the angle.
Let x be the measure of the angle. Then 180 – x is the measure of
its supplement, and 90 – x is the measure of its complement.
Supplement
is
180 – x
=
180 – x = 3.5(90 – x)
180 – x = 315 – 3.5x
180 + 2.5x = 315
2.5x = 135
x = 54
Complement 3.5 times smaller then supplement
3.5(90 – x)
So the measure of the angle is 54 degrees.
Example 4: The measure of the supplement of an angle is 60 less
than three times the measure of the complement of the angle.
Find the measure of the angle.
Let x be the measure of the angle. Then 180 – x is the measure of
its supplement, and 90 – x is the measure of its complement.
Supplement
is
180 – x
=
180 – x = 3(90 – x) – 60
180 – x = 270 – 3x – 60
180 – x = 210 – 3x
180 + 2x = 210
2x = 30
x = 15
60 less than three times the complement
3(90 – x) – 60
So the measure of the angle is 15 degrees.
Example 5: Find m<S if m<S is 20 more than four times its
supplement.
Let x = m<S. Then 180 – x is the measure of its supplement.
m<S
is
x
=
20 more than four times its supplement
x = 4(180 – x) + 20
x = 720 - 4x + 20
x = 740 – 4x
5x = 740
x = 148
So m<S is 148 degrees.
4(180 – x) + 20
Example 6: Refer to the
figure at the right.
C
D
B
F
A
A) Identify a pair of obtuse vertical angles.
<AFE and <BFD
B) Name a segment that is perpendicular to ray FC.
Ray AD
C) Which angle forms a linear pair with angle DFE?
<EFA or <DFB
D) Which angle is complementary to <CFB?
<BFA
E
Example 7: <N is a complement of <M, m<N = 8x – 6 and m<M
= 14x + 8. Find the value of x and m<M.
Since the two angles are complements, we know they add up
to 90.
m<N + m<M = 90
8x – 6 + 14x + 8 = 90
22x + 2 = 90
22x = 88
x=4
Plug 4 in for x in the equation for m<M.
m<M = 14x + 8
m<M = 14(4) + 8
m<M = 56 + 8
m<M = 65