1. 2.2 Angle Bisectors

2. Objectives
Bisect an angle.

3. Key Vocabulary
Angle Bisector

4. Angle Bisector
A ray that divides an angle into two congruent angles is called an angle bisector.
If AD bisects ∠BAC then ∠BAD is congruent to ∠CAD. B D A C
How do we know ∠BAD≅∠CAD? We label them congruent.

5. Labeling Angles
An arc that crosses 2 or more angles identifies the measure of the entire angle it crosses.
Matching arcs identify congruent angles in diagrams.
∠EFH≅∠HFG
∠GFJ≅∠JFK

6. Example 1 BD bisects ABC. Substitute 110° for mABC. Simplify.
BD bisects ABC, and mABC = 110°.
Find mABD and mDBC.
SOLUTION 2 1
(mABC)
mABD =
BD bisects ABC. 2 1 =
(110°)
Substitute 110° for mABC. =
55°
Simplify.
ABD and DBC are congruent, so mDBC = mABD.
ANSWER
So, mABD = 55°, and mDBC = 55°. 6

7. Your Turn: HK bisects GHJ. Find mGHK and mKHJ. ANSWER 26°; 26°1.
ANSWER
26°; 26° 2.
ANSWER
45°; 45° 3.
ANSWER
80.5°; 80.5°

8. Example 2 MP SOLUTION bisects LMN, and mLMP = 46°.
Find mPMN and mLMN. a.
Determine whether LMN is acute, right, obtuse, or straight. Explain. b.
SOLUTION a.
bisects LMN, so mLMP = mPMN . MP
You know that mLMP = 46°. Therefore, mPMN = 46°.
The measure of LMN is twice the measure of LMP.
mLMN = 2(mLMP) = 2(46°) = 92°
So, mPMN = 46°, and mLMN = 92°
LMN is obtuse because its measure is between 90° and 180°. b. 8

9. Your Turn:
QS bisects PQR. Find mSQP and mPQR. Then determine whether PQR is acute, right, obtuse, or straight. 1.
ANSWER
29°; 58°; acute 2.
ANSWER
45°; 90°; right 3.
ANSWER
60°; 120°; obtuse

10. Example 3
In the kite, DAB is bisected AC, and BCD is bisected by CA. Find mDAB and mBCD.
SOLUTION
2(mABC)
mDAB =
AC bisects DAB. =
2(45°)
Substitute 45° for mBAC. =
90°
Simplify.
2(mACB)
mBCD =
CA bisects BCD. =
2(27°)
Substitute 27° for mACB. =
54°
Simplify.
The measure of DAB is 90°, and the measure of BCD is 54°.
ANSWER 10

11. Your Turn: 1. KM bisects JKL. Find mJKM and mMKL. 2.
ANSWER
48°; 48° 2.
UV bisects WUT.
Find mWUV and mWUT.
ANSWER
60°; 120°

12. Example 4 RQ bisects PRS. Find the value of x. RQ bisects PRS.
SOLUTION
mPRQ =
mQRS
RQ bisects PRS.
Substitute given measures. =
85°
(6x + 1)°
Subtract 1 from each side. =
85 – 1
6x + 1 – 1
Simplify.
6x = 84
Divide each side by 6. 6x 6 –– = 84
Simplify.
x = 14
You can check your answer by substituting 14
for x.
mPRQ = (6x + 1)° = (6 · )° = (84 + 1)° = 85°
CHECK 12

13. Your Turn: BD bisects ABC. Find the value of x. 1. 2. ANSWER 43

14. Joke Time Why do gorillas have big nostrils?
Because they have big fingers.
Why are there so many Smiths in the phone book?
They all have phones.

15. Assignment
Section 2-2, pg : #1-21 odd, 29, 33