2.2 Angle Bisectors

Objectives Bisect an angle.

Key Vocabulary Angle Bisector

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.

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

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

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°

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

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

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

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°

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 · 14 + 1)° = (84 + 1)° = 85° CHECK 12

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

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.

Assignment Section 2-2, pg. 64-66: #1-21 odd, 29, 33