Question

Given: Circle O; Angle A intercepts Arc B C D ; Angle D intercepts Arc B C
Prove: Angle A is-congruent-to Angle D

Circle O is shown. Angles A and D intercept arc B C. Angles B and C intercept arc A D.

Two statements are missing reasons. What reason can be used to justify both statements 2 and 3?


Statements

Reasons
1. circle O; Angle A intercepts arc B C. Angle D intercects arc B C 1. given
2. Measure of angle A = one-half (measure of arc B C) 2. ?
3.
Measure of angle D = one-half (measure of arc B C)

3.
?

4.
Measure of angle A = measure of angle D

4. substitution property
5.
Angle A is-congruent-to Angle D

5. definition of congruent angles
inscribed angles theorem
third corollary to the inscribed angles theorem
central angle of a triangle has the same measure as its intercepted arc.
Angle formed by a tangent and a chord is half the measure of the intercepted arc.

Answer 1

Reason: the Inscribed Angle Theorem.

Because A and D are inscribed angles that intercept arc BC, each angle's measure equals one-half the measure of arc BC (Inscribed Angle Theorem). Thus statements 2 and 3 are both justified by the Inscribed Angle Theorem.