Question
Quadrilateral ABCD is inscribed in circle O below. Segment AC is a diameter of the circle.
image 39168155711a4430adc685d10c995ac7
What is the sum of the degree measures of
∠
A
B
C
and
∠
C
D
A
?
A.
90
°
B.
180
°
C.
270
°
D.
360
°
image 39168155711a4430adc685d10c995ac7
What is the sum of the degree measures of
∠
A
B
C
and
∠
C
D
A
?
A.
90
°
B.
180
°
C.
270
°
D.
360
°
Answers
GPT-4o mini
Since quadrilateral ABCD is inscribed in a circle and segment AC is a diameter of the circle, we can apply the inscribed angle theorem. According to this theorem, an angle inscribed in a semicircle is a right angle, which applies here because angles B and D are both inscribed in the semicircles formed by diameter AC.
Thus:
- Angle \( \angle ABC \) (which is angle B) is \( 90^\circ \) because it is subtended by the diameter AC.
- Angle \( \angle ADC \) (which is angle D) is also \( 90^\circ \) for the same reason.
Now, we need to find the sum of the degree measures of \( \angle ABC + \angle CDA \):
\[
\angle ABC + \angle CDA = 90^\circ + 90^\circ = 180^\circ.
\]
Therefore, the answer is \( 180^\circ \), which corresponds to option B.
Thus:
- Angle \( \angle ABC \) (which is angle B) is \( 90^\circ \) because it is subtended by the diameter AC.
- Angle \( \angle ADC \) (which is angle D) is also \( 90^\circ \) for the same reason.
Now, we need to find the sum of the degree measures of \( \angle ABC + \angle CDA \):
\[
\angle ABC + \angle CDA = 90^\circ + 90^\circ = 180^\circ.
\]
Therefore, the answer is \( 180^\circ \), which corresponds to option B.