1-4 Angles1-4 Angles1-4 Angles1-4 Angles

EXAMPLE 3 Find angle measures

oALGEBRA Given that m LKN =145 , find m LKM and m MKN.

SOLUTION

STEP 1

Write and solve an equation to find the value of x.

m LKN = m LKM + m MKN Angle Addition Postulate

Substitute angle measures.

145 = 6x + 7 Combine like terms.

Subtract 7 from each side.138 = 6x

Divide each side by 6.23 = x

145 = (2x + 10) + (4x – 3)o oo

EXAMPLE 3 Find angle measures

STEP 2

Evaluate the given expressions when x = 23.

m LKM = (2x + 10)° = (2 23 + 10)° = 56°

m MKN = (4x – 3)° = (4 23 – 3)° = 89°

So, m LKM = 56° and m MKN = 89°.ANSWER

GUIDED PRACTICE for Example 3

Find the indicated angle measures.

3. Given that KLM is straight angle, find m KLN and m NLM.

STEP 1

Write and solve an equation to find the value of x.

Straight angle

Substitute angle measures.

Combine like terms.

Subtract 2 from each side.

Divide each side by 14.

m KLM + m NLM = 180°

(10x – 5)° + (4x +3)°= 180°14x – 2 = 180

14x = 182

x = 13

SOLUTION

GUIDED PRACTICE for Example 3

STEP 2

Evaluate the given expressions when x = 13.

m KLM = (10x – 5)° = (10 13 – 5)° = 125°

m NLM = (4x + 3)° = (4 13 + 3)° = 55°

ANSWER m KLM = 125° m NLM = 55°

GUIDED PRACTICE for Example 3

4. Given that EFG is a right angle, find m EFH and m HFG.

STEP 1

Write and solve an equation to find the value of x.

Substitute angle measures.

Combine like terms.

Subtract 3 from each side.

Divide each side by 3.

m EFG + m HFGm EFG= = 90°

(2x + 2)° + (x +1)° = 90°3x + 3 = 90

3x = 87

x = 29

EFG is a right angle

SOLUTION

GUIDED PRACTICE for Example 3

STEP 2

Evaluate the given expressions when x = 29.

m EFH = (2x + 2)° = (2 29 +2)° = 60°

m HFG = (x + 1)° = (29 + 1)° = 30°

ANSWER m EFG = 60° m HFG = 30°

EXAMPLE 4 Identify congruent angles

The photograph shows some of the angles formed by the ropes in a trapeze apparatus. Identify the congruent angles. If m DEG = 157° ,what is m GKL?

Trapeze

SOLUTION There are two pairs of congruent angles:

DEF JKL and DEG GKL.~ ~

Because ∠DEG GKL, DEG = m GKL. So, m GKL = 157°.

~

GUIDED PRACTICE for Example 4

5. Identify all pairs of congruent angles in the diagram.

Use the diagram shown below.

There are two pairs of Congruent angles in the diagram.

T S and P Q.~ ~

SOLUTION

GUIDED PRACTICE for Example 4

6. In the diagram, m PQR = 130 , m QRS = 84, and m TSR = 121 . Find the other angle measures in the diagram.

o

o

o

PTS TSR~ = 121°

~ QRS QPT= 84°

Use the diagram shown at the right.

SOLUTION

Congruent angles

Congruent angles

SOLUTION

EXAMPLE 5 Double an angle measure

In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ.

o

By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~

So, m XYW = m WYZ, and you can write

M XYZ = m XYW + m WYZ = 18° + 18° = 36°.

GUIDED PRACTICE for Example 5

7. Angle MNP is a straight angle, and NQ bisects MNP. Draw MNP And NQ . Use arcs to mark the congruent angles in your diagram, and give

the angle measures of these congruent angles.

SOLUTION

GUIDED PRACTICE for Example 5

The solution is = 90°m MNQ = m PNQ

m MNQ + m PNQ = 180°

m MNQ + m PNQ

m MNQ + m MNQ = 180°

m MNQ2 = 180°

m MNQ = 90°

Angle addition postulate

Straight angle

m MNQ = m PNQ

Add

Divided each side by 2

EXAMPLE 1 Name angles

Name the three angles in the diagram.

WXY, or YXW

YXZ, or ZXY

WXZ, or ZXW

You should not name any of these angles X because all three angles have X as their vertex.

EXAMPLE 2 Measure and classify angles

Use the diagram to find the measure of the indicated angle. Then classify the angle.

a. KHJ b. GHK c. GHJ d. GHL

SOLUTION

A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.

EXAMPLE 2 Measure and classify angles

a. HJ is lined up with the 0 on the inner scale of the protractor. HK passes through 55 on the inner scale. So, m KHJ = 55 . It is an acute angle.

o

o

o

b. HG is lined up with the 0 on the outer scale and HK passes through 125 on the outer scale. So,m GHK = 125 . It is an obtuse angle.

o

o

o

c. m GHJ = 180. It is a straight angle.o

d. m GHL= 90. It is a right angle.o

GUIDED PRACTICE for Examples 1and 2

1. Name all the angles in the diagram at the right.Which angle is a right angle?

PQR , PQS, RQS . PQS is a right angle .

ANSWER

GUIDED PRACTICE for Examples 1and 2

2. Draw a pair of opposite rays. What type of angle do the rays form?

ANSWER Straight Angle