Download Inscribed Angles

NAME _____________________________________________ DATE ____________________________ PERIOD _____________
10-4 Study Guide and Intervention
Inscribed Angles
Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose
̂ is the intercepted arc for inscribed
sides contain chords of the circle. In ⨀G, minor arc 𝐷𝐹
angle ∠DEF.
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.
If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.
̂ = 90. Find m∠DEF.
Example: In ⨀G above, m 𝑫𝑭
∠DEF is an inscribed angle so its measure is half of the intercepted arc.
1
̂
m∠DEF = 2m𝐷𝐹
1
= 2(90) or 45
Exercises
Find each measure.
̂
1. m𝐴𝐶
̂
3. m𝑄𝑆𝑅
2. m∠N
ALGEBRA Find each measure.
4. m∠U
6. m∠A
5. m∠T
7. m∠C
Chapter 10
23
Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
10-4 Study Guide and Intervention (continued)
Inscribed Angles
Angles of Inscribed Polygons An inscribed polygon is one
whose sides are chords of a circle and whose vertices are points on the
circle. Inscribed polygons have several properties.
• An inscribed angle of a triangle intercepts a diameter or semicircle
if and only if the angle is a right angle.
• If a quadrilateral is inscribed in a circle, then its opposite angles are
supplementary.
̂ is a semicircle, then m∠BCD = 90.
If 𝐵𝐶𝐷
For inscribed quadrilateral ABCD,
m∠A + m∠C = 180 and
m∠ABC + m∠ADC = 180.
Example: Find m∠ K.
̂ ≅ 𝐾𝑀
̂ , so KL = KM. The triangle is an isosceles triangle, therefore m∠L = m∠M = 3x + 5.
𝐾𝐿
m∠L + m∠M + m∠K = 180
(3x + 5) + (3x + 5) + (5x + 5) = 180
11x + 15 = 180
11x = 165
x = 15
Angle Sum Theorem
Substitution
Simplify.
Subtract 15 from each side.
Divide each side by 11.
So, m∠K = 5(15) + 5 = 80.
Exercises
ALGEBRA Find each measure.
1. x
3. x
2. m∠W
4. m∠T
5. m∠R
7. m∠W
6. m∠S
8. m∠X
Chapter 10
24
Glencoe Geometry