Angle Addition and
Bisector
Homework: Textbook pg 19 #9-11;
pg 20 #26-29; pg 25 #1-7, 15-16
PreAP: Same as above and
Textbook pg 26 #19-23
Example 1
Angle Name
Angle Measure
∠𝑀𝐿𝑁
60
∠𝑀𝐿𝑂
90
∠𝑀𝐿𝑃
130
∠𝑀𝐿𝑄
180
∠𝑁𝐿O
30
∠𝑁𝐿P
70
∠𝑁𝐿𝑄
120
∠𝑂𝐿𝑃
40
∠𝑂𝐿𝑄
90
∠𝑃𝐿𝑄
50
mAOB = 4x  1; mBOC = 2x + 15; mAOC = 8x + 8
Example 2
Work
4𝑥 − 1 + 2𝑥 + 15 = 8𝑥 + 8
6𝑥 + 14 = 8𝑥 + 8
14 = 2𝑥 + 8
6 = 2𝑥
3 = 𝑥 or 𝑥 = 3
𝑚∠𝐴𝑂𝐵 = 4 3 − 1 = 11
𝑚∠𝐵𝑂𝐶 = 2 3 + 15 = 21
𝑚∠𝐴𝑂𝐶 = 8 3 + 8 = 32
Example 3
mCOD = 8x + 13; mBOC = 3x  10; mBOD = 12x  6
Work
8𝑥 + 13 + 3𝑥 − 10 = 12𝑥 − 6
11𝑥 + 3 = 12𝑥 − 6
3=𝑥−6
9 = 𝑥 or 𝑥 = 9
𝑚∠𝐶𝑂𝐷 = 8 9 + 13 = 85
𝑚∠𝐵𝑂𝐶 = 3 9 − 10 = 17
𝑚∠𝐷𝑂𝐵 = 12 9 − 6 = 102
If mRZT = 110, mRZS = 3s, and mTZS = 8s, what are
mRZS and mTZS?
Work
3𝑠 + 8𝑠 = 110
11𝑠 = 110
𝑠 = 10
𝑚∠𝑅𝑍𝑆 = 3 10 = 30
𝑚∠𝑇𝑍𝑆 = 8 10 = 80
Example 4
mOZP = 4r + 2, mPZQ = 5r  12, and mOZQ = 125. What are
mOZP and mPZQ?
Example 5
Work
4𝑟 + 2 + 5𝑟 − 12 = 125
9𝑟 − 10 = 125
9𝑟 = 135
𝑟 = 15
𝑚∠𝑂𝑍𝑃 = 4 15 + 2 = 62
𝑚∠𝑃𝑍𝑄 = 5 15 − 12 = 63
QS bisects PQR. Solve for x and find mPQR
mPQR = 3x  12; mPQS = 30
Work
Bisects means congruent angles
2 30 = 3𝑥 − 12
60 = 3𝑥 − 12
72 = 3𝑥
24 = 𝑥 or x = 24
𝑚∠𝑃𝑄𝑅 = 3 24 − 12 = 60
Example 6
QS bisects PQR. Solve for x and find mPQR
mPQS = 2x + 10; mSQR = 5x  17
Work
Bisects means congruent angles
2𝑥 + 10 = 5𝑥 − 17
10 = 3𝑥 − 17
27 = 3𝑥
9 = 𝑥 or x = 9
𝑚∠𝑃𝑄𝑆 = 2 9 + 10 = 28
𝑚∠𝑃𝑄𝑅 = 2 28 = 56
Example 7
QS bisects PQR. Solve for x and find mPQR
mPQS = 3x; mSQR = 5x  20
Work
Bisects means congruent angles
3𝑥 = 5𝑥 − 20
−2𝑥 = −20
𝑥 = 10
𝑚∠𝑃𝑄𝑆 = 3 10 = 30
𝑚∠𝑃𝑄𝑅 = 2 30 = 60
Example 8
QS bisects PQR. Solve for x and find mPQR
mPQS = 2x + 1; mRQS = 4x  15
Work
Bisects means congruent angles
2𝑥 + 1 = 4𝑥 − 15
1 = 2𝑥 − 15
16 = 2𝑥
8 = 𝑥 or x = 8
𝑚∠𝑃𝑄𝑆 = 2 8 + 1 = 17
𝑚∠𝑃𝑄𝑅 = 2 17 = 34
Example 9