Answer:
# A circumscribed angle is created by two intersecting tangent segments ⇒ true (1st answer)
# The measure of a central angle will be twice the measure of an inscribed angle that intercepts the same arc ⇒ true (3rd answer)
# The measure of a central angle will be equal to the measure of an inscribed angle when the arc intercepted by the inscribed angle is twice as large as the arc intercepted by the central angle ⇒ true (6th answer)
Step-by-step explanation:
* Lets revise the types of angles in a circle
- A circumscribed angle is the angle made by two intersecting
tangent lines to a circle (it's out side the circle)
- Its measure is half the difference of the measures of the two
intercepted arcs
- Ex:
∵ AB and AC are tangent to circle M at B and C
∴ ∠A is a circumscribed angle
∴ m∠A = 1/2(m major arc BC - m minor arc BC)
- An inscribed angle is an angle formed by two chords in a circle
which have a common endpoint, this common endpoint is the
vertex of it
- Its measure is half the measure of the intercepted arc
Ex:
∵ XY and XZ are two chords in circle M
∴ ∠YXZ is an inscribed angle subtended by arc YZ
∴ m∠YXZ = 1/2 (m arc YZ)
- A central angle is an angle with endpoints located on the
circumference of the circle and its vertex is the center of the circle
- Its measure is the measure of the intercepted arc
- Ex:
∵ MA and MB are two radii of circle M
∴ ∠AMB is a central angle subtended by the opposite arc AB
∴ m∠AMB = m of arc AB
- The measure of an inscribed angle is half the measure of the
central angle which subtended by the same arc
- Ex:
∵ ∠ABC is an inscribed angle in circle M subtended by arc AC
∵ ∠AMC is a central angle subtended by arc AC
∴ m∠ABC = 1/2 m∠AMC
∴ m∠AMC = 2 m∠ABC
* Lets solve the problem
- From the facts above:
# A circumscribed angle is created by two intersecting tangent
segments ⇒ true
# The measure of a central angle will be twice the measure of an
inscribed angle that intercepts the same arc ⇒ true
- Lets prove the last statement
∵ AMC is a central angle of circle M subtended by arc AC
∴ m∠AMC = m of arc AC ⇒ (1)
∵ XYZ is an inscribed angle of circle M subtended by arc XZ
∴ m∠XYZ = 1/2 m of arc XZ
∵ m of arc XZ is twice m of arc AC
∴ m∠XYZ = m of arc AC ⇒ (2)
- From (1) and (2)
∴ m∠AMC = m∠XYZ
∴ The statement down is true
# The measure of a central angle will be equal to the measure of
an inscribed angle when the arc intercepted by the inscribed
angle is twice as large as the arc intercepted by the central
angle ⇒ true
