Question

a sector of a circle is a section shaped like a piece of pie, bounded by two radii and an arc. which of the following conditions would always be satisfied if two sectors of the same circle were congruent? select all that apply. a. the arcs must be complementary b. the segments joining the radius and arc must be identical c. the central angles must have the same measure d. the central angles must be complementary e. the central angles must be supplementary f. the segments joining endpoints of each arc are congruent g. the arcs must be supplementary h. the arcs must have the same measure

Explanation:

Step1: Recall congruent - sector definition

Two sectors of the same circle are congruent if they have the same central - angle measure and the same arc - length measure. Also, the segments joining the endpoints of each arc (the chords) are congruent.

Step2: Analyze option A

Complementary arcs add up to 90°, not a requirement for congruent sectors.

Step3: Analyze option B

The segments joining the radius and arc are part of the sector's structure. For congruent sectors of the same circle, they are identical.

Step4: Analyze option C

Same - measure central angles are a key condition for congruent sectors.

Step5: Analyze option D

Complementary central angles add up to 90°, not a requirement for congruent sectors.

Step6: Analyze option E

Supplementary central angles add up to 180°, not a requirement for congruent sectors.

Step7: Analyze option F

The segments joining endpoints of each arc (chords) are congruent for congruent sectors.

Step8: Analyze option G

Supplementary arcs add up to 180°, not a requirement for congruent sectors.

Step9: Analyze option H

Same - measure arcs are a key condition for congruent sectors.

Answer:

B. The segments joining the radius and arc must be identical,
C. The central angles must have the same measure,
F. The segments joining endpoints of each arc are congruent,
H. The arcs must have the same measure