Upload
georgiana-flynn
View
220
Download
0
Tags:
Embed Size (px)
Citation preview
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