Download Geometry 10-4 Inscribed Angles

Geometry 10-4 Inscribed Angles
A. Inscribed angles
1. An inscribed angle is an angle that has its vertex and its sides contained
in the chords of the circle.
2. Theorem 10-5 - Inscribed Angle Theorem - If an angle is inscribed in
a circle, then the measure of the angle equals one-half the measure of its
intercepted arc (or the measure of the intercepted
arc is twice the measure of the inscribed angle).
Ex: m ∠ ABC = ½ mAC or 2(m ∠ ABC) = mAC
Ex 1: In circle F , mWX = 20, mXY = 40, mUZ = 108, and mUW = mYZ .
Find the measures of the numbered angles.
3. Theorem 10-6 - If two inscribed angles of a circle (or congruent circles)
intercept congruent arcs or the same arc, then the angles are congruent.
Abbreviations:
Inscribed ∠ ’s of ≅ arcs are ≅ .
Inscribed ∠ ’s of same arc are ≅ .
∠ DAC ≅ ∠ ______
B. Angles of Inscribed Polygons
1. Theorem 10-7 - If an inscribed angle intercepts a
semicircle, the angle is a right angle.
ADC is a semicircle so m ∠ ABC = 90°
Ex 3: Triangles TVU and TSU are inscribed in
Circle P with VU ≅ SU . Find the measure of each
numbered angle if m ∠ 2 = x + 9 and
m ∠ 4 = 2x + 6 .
Ex 4: Quadrilateral QRST is inscribed in Circle M.
If m ∠ Q = 87 and m ∠ R = 102. Find m ∠ S and m ∠ T.
R
Q
M
2. Theorem 10-8 - If a quadrilateral is inscribed in a
circle, then its opposite angles are supplementary.
HW: Geometry 10-4 p. 549-551
8-10, 13-16, 18-20, 22-29, 48-52
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