Question

Which statements are true regarding the relationships between central, inscribed, and circumscribed angles of a circle? Check all that apply.

A circumscribed angle is created by two intersecting tangent segments.

A central angle is created by two intersecting chords that are not a diameter.

The measure of a central angle will be twice the measure of an inscribed angle that intercepts the same arc.

The measure of a central angle will be half the measure of an inscribed angle that intercepts the same arc.

The measures of a central angle and circumscribed angle that intercept the same arc will sum to 90°.

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.

Answer 1

# 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)

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

Answer 2

A circumscribed angle is created by two intersecting tangent segments.

The measure of a central angle will be twice the measure of an inscribed angle that intercepts the same arc.

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.

Explanation:

We can choose the correct answers by studying their definitions.

A circumscribed angle is defined by two tangent segments, which have an external common point, to their tangent point, as you can see in the image attached, thats why the first answer is correct.

The third answer belongs to a geometrical conjecture, which says: ''In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc''. This conjecture stablished a relation between a central angle and an inscribed angle. Textually, it says that the inscribed is half of the central, which is the same if we say, the central is twice the inscribed.

The last answer is a variation of the conjecture we used before. In this case, if the arc is twice, then inscribed and central are equal, this is deducted from that conjecture, only considering the opposite case, which angles are equal and arcs don't.