Inscribed Angles and Their Properties

This video tutorial covers inscribed angles, their properties, and related theorems. It explains the inscribed angle theorem, which states that the measure of an inscribed angle is half the measure of the intercepted arc. The video also discusses three corollaries: congruent angles intercepting the same arc, angles inscribed in a semicircle being right angles, and opposite angles of an inscribed quadrilateral being supplementary. Additionally, the tangent and chord theorem is introduced, which states that the measure of an angle formed by a tangent and a chord is half the intercepted arc. The tutorial includes problem-solving examples to find missing angle measures.

What is an inscribed angle?

• An angle with its vertex outside the circle

• An angle with its vertex at the center of the circle

• An angle with its vertex inside the circle but not at the center

• An angle with its vertex on the circle and sides containing chords

According to the inscribed angle theorem, the measure of an inscribed angle is:

• One-third the measure of the intercepted arc

• Half the measure of the intercepted arc

• Twice the measure of the intercepted arc

• Equal to the measure of the intercepted arc

What does the first corollary of the inscribed angle theorem state?

• Inscribed angles intercepting the same arc are congruent

• Opposite angles of an inscribed quadrilateral are supplementary

• An inscribed angle in a semicircle is a right angle

• The measure of an angle formed by a tangent and a chord is half the intercepted arc

What is the measure of an angle inscribed in a semicircle?

• 180 degrees

• 90 degrees

• 60 degrees

• 45 degrees

What is the relationship between opposite angles of a quadrilateral inscribed in a circle?

• They are complementary

• They are equal

• They are supplementary

• They are congruent

If an inscribed angle intercepts arcs of 60 and 80 degrees, what is the measure of the angle?

• 110 degrees

• 90 degrees

• 120 degrees

• 70 degrees

What does theorem 12-12 state about angles formed by a tangent and a chord?

• The angle is equal to the intercepted arc

• The angle is twice the intercepted arc

• The angle is half the intercepted arc

• The angle is one-third the intercepted arc

In the diagram, if arc PMQ is 212 degrees, what is the measure of angle PQR?

• 148 degrees

• 74 degrees

• 106 degrees

• 212 degrees

What is the measure of angle Y if it intercepts a semicircle and another angle of 35 degrees?

• 75 degrees

• 65 degrees

• 55 degrees

• 45 degrees

What is the measure of angle X if it shares the same intercepted arc as angle JQL?

• 25 degrees

• 55 degrees

• 45 degrees

• 35 degrees