Page 155 - Math Course 3 (Book 2)

Circles: Angles and Arcs

Measures of Arcs
In ⊙P, m∠NPM = 46, PL bisects ∠KPM, and OP⟂KN.

Examples Find mJKO.

m∠NPM = 46
L Vertical angles are
K m∠KPJ = m∠NPM congruent.

m∠KPJ = 46 Substitution.
P M

J m∠KPJ + m∠JPO = 90 ∠KPO is a right angle
46 + m∠JPO = 90 Substitution.
N
Subtract 46 from each
O m∠JPO = 44 side.

In ⊙P, m∠NPM = 46, PL bisects ∠KPM, and OP⟂KN. m∠JPO = mJO = 44
Find mOK.
mJO + mJKO = 360

OK is a minor arc, so mOK = mKPO 44 + mJKO = 360 Substitution.
KON is a semicircle Subtract 44 from
mJKO = 316
each side.
mON = m∠NPO

Answer 316
= 90 ∠NPO is a right triangle
mKON = mOK + mON Arc Addition Postulate

180 = mOK + 90 Substitution Your Turn!

90 = mOK Subtract 90 from each side.
Measures of Central Angles

Answer 90 ALGEBRA
Refer to ⊙Z. Find m∠CZD
(5x – 5)°
A B
In ⊙P, m∠NPM = 46, PL bisects ∠KPM, and OP⟂KN. A. 9
B. 21 75° C
Find mLM. C. 65
D. 30
1 Z (7x + 2)°
mLM = KM since PL bisects ∠KPM
2
KMN is a semicircle.

E D
mKM + MN = mKMN Arc Addition Postulate
mKM + 46 = 180 mMN = m∠NPM = 46

Subtract 46 from each
mKM = 134 side.
1 Answer
mLM = (134)
2
or 67

Answer 67

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