UNIT 1.4-1.5 EXPLORING UNIT 1.4-1.5 EXPLORING ANGLES ANGLES
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UNIT 1.4-1.5 EXPLORING UNIT 1.4-1.5 EXPLORING ANGLESANGLES
Two different Rays with the Two different Rays with the same Endpointsame Endpoint
The Rays of the AngleThe Rays of the Angle
The Endpoint of the AngleThe Endpoint of the Angle
An Angle between 0° and An Angle between 0° and 90 °.90 °.
An Angle that equals 90 °An Angle that equals 90 °
An Angle between 90 ° and An Angle between 90 ° and 180 °180 °
An Angle that equals 180 °An Angle that equals 180 °
The measure of an angle is The measure of an angle is equal to the absolute value equal to the absolute value of the difference between of the difference between values of the rays on the values of the rays on the
protractor.protractor.
If P is in the interior of If P is in the interior of <RST, then m<RST = <RST, then m<RST =
m<RSP + m<PSTm<RSP + m<PST
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 EXAMPLE 22 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.
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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.
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c. m GHJ = 180. It is a straight angle.
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d. m GHL= 90. It is a right angle.
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GUIDED PRACTICE for Examples 1and 2
1. Name all the angles in the diagram. 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
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 a straight angle, find m KLN and m NLM.
ANSWER 125°, 55°
GUIDED PRACTICE for Example 3
4. Given that EFG is a right angle, find m EFH and m HFG.
ANSWER 60°, 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°.
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GUIDED PRACTICE for Example 4
5. Identify all pairs of congruent angles in the diagram.
Use the diagram shown.
T and S, P and R.ANSWER
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.
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Use the diagram shown.
m PTS = 121°, m QPT = 84°ANSWER
SOLUTION
EXAMPLE 5 Double an angle measure
In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ.
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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.
90°ANSWER
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