WARM-UPRefer to the figure below.

A CD

G

B

3

21

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.

WARM-UPRefer to the figure below.

A CD

G

B

3

21

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

WARM-UPRefer to the figure below.

A CD

G

B

3

21

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<ABC3x + 4 + 2x – 3 = 715x + 1 = 715x = 70x = 14Plug 14 in for x in the equation for m<ABD.m<ABD = 3x + 4m<ABD = 3(14) + 4m<ABD = 42 + 4m<ABD = 46

5) Suppose MN bisects <RMQ, m<RMN = x + 2, and m<NMQ = 2x – 3. Find the value of x and m<RMQ.

M

N

Q

R

Since MN bisects <RMQ, we know that m<RMN = m<NMQ.m<RMN = m<NMQx + 2 = 2x – 32 = x – 35 = xPlug 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 + 4m<RMQ = 2(5) + 4m<RMQ = 10 + 4m<RMQ = 14

CHAPTER 1SECTION 7

Angle Relationships

VOCABULARYPerpendicular 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.

 

21

3

4

VOCABULARY CONT.Linear Pair- Adjacent angles whose noncommon sides are opposite rays.( <1 and <2 form a linear pair)

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) 

6

5

1 2

43

Example 1: Use the figure below to find the value of x and m<JIH.

K

G J

I

16x - 20

H13x + 7

Since <GIJ and <KIH are vertical angles, we know they are congruent.m<GIJ = m<KIH16x – 20 = 13x + 73x – 20 = 73x = 27x = 9Since <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 = 18013x + 7 + m<JIH = 18013(9) + 7 + m<JIH = 180117 + 7 + m<JIH = 180124 + m<JIH = 180m<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.

K

GJ

I

9x - 4

H

4x - 11

Since <GIJ and <JIH are supplementary and form a linear pair, their measures add up to 180.m<GIJ + m<JIH = 1809x – 4 + 4x – 11 = 18013x – 15 = 18013x = 195x = 15Since <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 – 4m<GIJ = 9(15) - 4m<GIJ = 135 – 4m<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.

180 – x = 3.5(90 – x) 180 – x = 3.5(90 – x)180 – x = 315 – 3.5x180 + 2.5x = 315 2.5x = 135x = 54

So the measure of the angle is 54 degrees.

Supplement

is Complement 3.5 times smaller then supplement

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.

180 – x = 3(90 – x) – 60180 – x = 3(90 – x) – 60180 – x = 270 – 3x – 60180 – x = 210 – 3x180 + 2x = 2102x = 30x = 15

So the measure of the angle is 15 degrees.

Supplement

is 60 less than three times the complement

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.

x = 4(180 – x) + 20

x = 4(180 – x) + 20x = 720 - 4x + 20x = 740 – 4x5x = 740x = 148

So m<S is 148 degrees.

m<S is 20 more than four times its supplement

Example 6: Refer to the figure at the right.

B

C D

EF

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

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 = 908x – 6 + 14x + 8 = 9022x + 2 = 9022x = 88x = 4

Plug 4 in for x in the equation for m<M.m<M = 14x + 8m<M = 14(4) + 8m<M = 56 + 8m<M = 65