Transcript 2.2 Angle Bisectors

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, and mABC = 110°.
Find mABD and mDBC.
SOLUTION
mABD
=
=
1
2
1
2
(mABC)
BD bisects ABC.
(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°.
Your Turn:
HK bisects GHJ. Find mGHK and mKHJ.
1.
ANSWER
26°; 26°
ANSWER
45°; 45°
ANSWER
80.5°; 80.5°
2.
3.
Example 2
MP bisects LMN, and mLMP = 46°.
a. Find mPMN and mLMN.
b. Determine whether LMN is acute, right, obtuse, or
straight. Explain.
SOLUTION
a. MP bisects LMN, so mLMP = mPMN .
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°
b. LMN is obtuse because its measure is between 90° and
180°.
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
ANSWER
45°; 90°; right
2.
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
mDAB
= 2(mABC)
= 2(45°)
mBCD
AC bisects DAB.
Substitute 45° for mBAC.
= 90°
Simplify.
= 2(mACB)
CA bisects BCD.
= 2(27°)
Substitute 27° for mACB.
= 54°
ANSWER
Simplify.
The measure of DAB is 90°, and the
measure of BCD is 54°.
Your Turn:
1. KM bisects JKL.
Find mJKM and mMKL.
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.
SOLUTION
mPRQ
= mQRS
RQ bisects PRS.
(6x + 1)°
= 85°
Substitute given measures.
= 85 – 1
Subtract 1 from each side.
6x + 1 – 1
6x = 84
Simplify.
84
6x
–– = ––
6
6
Divide each side by 6.
Simplify.
x = 14
CHECK
You can check your answer by substituting 14
for x.
mPRQ = (6x + 1)° = (6 · 14 + 1)° = (84 + 1)° = 85°
Your Turn:
BD bisects ABC. Find the value of x.
1.
ANSWER
43
ANSWER
3
2.
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