Question

The figure shows circle C with inscribed quadrilateral J K L M.





Question
Which statements are sufficient to know that all four angles of J K L M ,begin emphasis,must,end emphasis, be right angles?



Choose ,begin emphasis,all,end emphasis, sufficient statements.

Answer options with 6 options
A.
angle J is congruent to angle L

B.
angle J and angle K are right angles

C.
the measure of angle J plus the measure of angle L equals 180° and the measure of angle K plus the measure of angle M equals 180°

D.
arc J K L and arc L M J are semicircles

E.
arc J K L and arc K L M are semicircles

F.
arc L M J is a semicircle and the measure of angle J plus the measure of angle K equals 180°

Answer 1

B, E, and F.

Brief justifications:
- B: If ∠J and ∠K are right, then chord KM and chord JL are diameters, so each inscribed angle subtending those diameters is 90°, giving all four right angles.
- E: Arc JKL semicircle ⇒ JL is a diameter (so ∠K and ∠M = 90°). Arc KLM semicircle ⇒ KM is a diameter (so ∠J and ∠L = 90°). Hence all four are 90°.
- F: Arc LMJ semicircle ⇒ JL is a diameter, so ∠K = 90°. Given ∠J + ∠K = 180° and ∠K = 90°, ∠J = 90°. Then ∠L (opposite ∠J in the cyclic quadrilateral) = 90° as well, so all four are right.