sec 3.3 angle addition postulate & angle bisector
TRANSCRIPT
• Find the measure of an angle by using Angle Addition Postulate.
• Find the measure of an angle by using definition of Angle Bisector.
Objective: What we’ll learn…
Angle Addition Postulate
First, let’s recall some previous information from last week….
We discussed the Segment Addition Postulate, which stated that we could add the lengths of adjacent segments together to get the length of an entire segment.
For example:
JK + KL = JL
If you know that JK = 7 and KL = 4, then you can conclude that JL = 11.
The Angle Addition Postulate is very similar, yet applies to angles. It allows us to add the measures of adjacent angles together to find the measure of a bigger angle…
J K L
Postulate 2-2 Segment Addition Postulate
If Q is between P and R, then PQ + QR = PR.
If PQ +QR = PR, then Q is between P and R.
P Q R
2x 4x + 6
PQ = 2x QR = 4x + 6 PR = 60
Use the Segment Addition Postulate find the measure of PQ and QR.
PQ + QR = PR (Segment Addition)
2x + 4x + 6 = 60 6x + 6 = 60 6x = 54 x =9 PQ = 2x = 2(9) = 18 QR =4x + 6 = 4(9) + 6 = 42
Step 1:
Step 2:
Step 3:
Step 4:
Steps
1. Draw and label the Line Segment.2. Set up the Segment Addition/Congruence
Postulate.3. Set up/Solve equation.4. Calculate each of the line segments.
Angle Addition Postulate
5065
AB
CO
If B lies on the interior of AOC, then mAOB + mBOC = mAOC.
mAOC = 115
Slide 2
Example 1:
This is a special example, because the two adjacent angles together create a straight angle.
Predict what mABD + mDBC equals.
ABC is a straight angle, therefore mABC = 180.
mABD + mDBC = mABC
mABD + mDBC = 180
So, if mABD = 134, then mDBC = ______
A B C
D
134°
46
G
H J
K
Given: mGHK = 95 mGHJ = 114.
Find: mKHJ.
The Angle Addition Postulate tells us:mGHK + mKHJ = mGHJ
95 + mKHJ = 114
mKHJ = 19.
95
114
19
Plug in what you
know.
Solve.
46°
Example 2: Slide 3
Algebra Connection
Slide 4R
S T
V
Given:mRSV = x + 5mVST = 3x - 9mRST = 68
Find x.
mRSV + mVST = mRST
x + 5 + 3x – 9 = 68
4x- 4 = 68
4x = 72
x = 18
Set up an equation using the Angle Addition Postulate.
Plug in what you
know.Solve.
Extension: Now that you know x = 18, find mRSV and mVST.
mRSV = x + 5mRSV = 18 + 5 = 23
mVST = 3x - 9mVST = 3(18) – 9 = 45
Check:mRSV + mVST = mRST23 + 45 = 68
Algebra Connection
Slide 5
B
QD
CmBQC = x – 7 mCQD = 2x – 1 mBQD = 2x + 34
Find x, mBQC, mCQD, mBQD.
mBQC + mCQD = mBQD
3x – 8 = 2x + 34
x – 7 + 2x – 1 = 2x + 34
x – 8 = 34
x = 42
mBQC = 35
mCQD = 83
mBQD = 118
x = 42
mBQC = x – 7mBQC = 42 – 7 = 35
mCQD = 2x – 1mCQD = 2(42) – 1 = 83
mBQD = 2x + 34mBQD = 2(42) + 34 = 118
Check:mBQC + mCQD = mBQD35 + 83 = 118
EXAMPLE 3 Find angle measures
oALGEBRA Given that m LKN =145 , find m LKM and m MKN.
So, m LKM = 56° and m MKN = 89°.ANSWER
EXAMPLE 3 Find angle measures
oALGEBRA Given that m LKN =145 , find m LKM and m MKN.
So, m LKM = 56° and m MKN = 89°.ANSWER
EXAMPLE 3 Find angle measures
oALGEBRA Given that m LKN =145 , find m LKM and m MKN.
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°
Congruent Angles
Identify all pairs of congruent angles in the diagram.
T and S, P and R.ANSWER
In the diagram, m∠Q = 130° , m∠R = 84°, and m ∠ S = 121° . Find the other angle measures in the diagram.
m T = 121°, m P = 84°ANSWER
Two angles are congruent if they have the same measure.
Congruent angles in a diagram are marked by matching arcs at the vertices .
Angle Bisecotrs
In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ. o
m XYZ = m XYW + m WYZ = 18° + 18° = 36°.
An angle bisector is a ray that divides an angle into two congruent angles.
EXAMPLE 3 Animated Solution – Click to see steps and reasons.
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
3.3 Angle Bisector3.3 Angle Bisector
• A ray that divides an angle into 2 congruent adjacent angles.
BD is an angle bisector. bisector of <ABC.B
A
C
D
Example 1 Find Angle Measures
2
1= (110°) Substitute 110° for mABC.
= 55° Simplify.
ABD and DBC are congruent, so mDBC = mABD.
ANSWER So, mABD = 55°, and mDBC = 55°.
SOLUTION
2
1(mABC)mABD = BD bisects ABC.
BD bisects ABC, and mABC = 110°.Find mABD and mDBC.
You know that mLMP = 46°. Therefore, mPMN = 46°.
The measure of LMN is twice the measure of LMP.
mLMN = 2(mLMP) = 2(46°) = 92°
So, mPMN = 46°, and mLMN = 92°
Example 2 Find Angle Measures and Classify an Angle
LMN is obtuse because its measure is between 90° and 180°.b.
Find mPMN and mLMN.a.
Determine whether LMN is acute, right, obtuse, or straight. Explain.
b.
bisects LMN, and mLMP = 46°.MP
SOLUTION
a. bisects LMN, so mLMP = mPMN .MP
Checkpoint Find Angle Measures
ANSWER 26°; 26°
ANSWER 80.5°; 80.5°
ANSWER 45°; 45°
1.
2.
3.
HK bisects GHJ. Find mGHK and mKHJ.
Checkpoint Find Angle Measures and Classify an Angle
ANSWER 29°; 58°; acute
ANSWER 45°; 90°; right
ANSWER 60°; 120°; obtuse
5.
6.
4.
QS bisects PQR. Find mSQP and mPQR. Then determine whether PQR is acute, right, obtuse, or straight.
Example 3 Real Life
= 2(45°) Substitute 45° for mBAC.
= 90° Simplify.
Substitute 27° for mACB.= 2(27°)
= 54° Simplify.
The measure of DAB is 90°, and the measure of BCD is 54°.
ANSWER
In the kite, DAB is bisected AC, and BCD is bisected by CA. Find mDAB and mBCD.
2(mACB)mBCD = CA bisects BCD.
SOLUTION
2(mABC)mDAB = AC bisects DAB.
Checkpoint Real Life
ANSWER 48°; 48°
ANSWER 60°; 120°
7. KM bisects JKL.Find mJKM and mMKL.
8. UV bisects WUT.Find mWUV and mWUT.
Construct the bisector of an angleusing a compass and straight edge
• Using the vertex O as a center, draw an arc to meet the arms of the angle (at X and Y).• Using X as a center and the same radius, draw a new arc.• Using Y as center and the same radius, draw an overlapping arc.• Mark the point where the arcs meet.• The bisector is the line from O to this point.
Y
X
O
A
B
E
Solve for x.
x+40o
3x-20o
* If they are congruent, set them equal to each other,
then solve!
x+40=3x-20
40=2x-20
60=2x
30=x
Example 4 Use Algebra with Angle Measures
Simplify.x = 14
Substitute given measures.= 85°(6x + 1)°
Subtract 1 from each side.= 85 – 16x + 1 – 1
Simplify.6x = 84
Divide each side by 6.6x
6–– =
84
6––
You can check your answer by substituting 14 for x. mPRQ = (6x + 1)° = (6 · 14 + 1)° = (84 + 1)° = 85°
CHECK
SOLUTION
mPRQ = mQRS RQ bisects PRS.
RQ bisects PRS. Find the value of x.