Page 169 - Math Course 3 (Book 2)

Inscribed Angles

Angles of an Inscribed Triangle
Let’s Begin
Example

Inscribed Arcs and Probability ALGEBRA

Example Triangles TVU and TSU are inscribed in ⊙P with
VU ≅ SU. Find the measure of each numbered
angle if m∠2 = x + 9 and m∠4 = 2x + 6.
PROBABILITY U
Points M and N are on a circle so that mMN = 72. V
Suppose point L is randomly located on the same 3 4
circle so that it does not coincide with M or N. What
is the probability that m∠MLN = 144?
P S
If m∠MLN = 144, the angle measure would be twice
the arc measure. Then, inscribed ∠MLN must inter- 1
2
cept MBN, so L must lie on minor arc MN. Draw a
figure and label any information you know. T

mMN = 72° ΔUVT and ΔUST are right triangles. m∠1 = m∠2
since they intercept congruent arcs. Then the third
angles of the triangles are also congruent, so
M L m∠3 = m∠4.

144°
N m∠2 + m∠4 + m∠S = 180 Angle Sum Theorem

(x + 9) + (2x + 6) + 90 = 180 m∠2 = x + 9,
m∠4 = 2x + 6,
36° m∠S = 90

3x + 105 = 180 Simplify
B
3x = 75 Subtract 105 from
The probability that m∠MLN = 144 is the same as each side.
the probability of L being contained in MN. x = 25 Divide each side by 3.

Use the value of x to find the measures of
The probability that L is located
Answer 72 1 ∠1, ∠2, ∠3 and ∠4.
on MN is or
360 5
m∠2 = x + 9 Given m∠4 = 2x +6 Given

= 25 + 9 x = 25 = 2(25) + 6 x = 25

= 34 = 56

m∠1 = m∠2 =34 m∠3 = m∠4 =56

m∠1 = 43; m∠2 = 43; m∠3 = 56;
Answer
m∠4 = 56;

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