Question

AOC and BOD are diameters of a circle, center O. Prove that triangle ABD and triangle DCA are congruent by RHS.

A) ADB and ADC share the same side DA.
B) Both triangles share the same side AD and congruent hypotenuses.
C) The triangles are congruent because they have the same base AB and altitude DC.
D) Triangle ADB is congruent to triangle DCA because all sides and angles are equal.

Answer 1

To prove that triangle ABD and triangle DCA are congruent by RHS, we need to show that they have a right angle, congruent hypotenuses, and a shared side AD. The correct option is B) Both triangles share the same side AD and congruent hypotenuses.

Step-by-step explanation:

In order to prove that triangle ABD and triangle DCA are congruent by RHS, we need to show that they have a right angle, a congruent hypotenuse, and a congruent side.

A) Triangle ADB and triangle DCA share the same side AD, but this alone is not enough to prove congruence.

B)Both triangles share the same side AD and congruent hypotenuses. This is correct because AOC and BOD are diameters, so AD is a diameter as well, making AD a congruent hypotenuse in both triangles.

C) The triangles are not congruent because they do not have the same base AB. The base of triangle ABD is AB and the base of triangle DCA is DC.

D) Triangle ADB is not congruent to triangle DCA because they don't have equal angles. Triangle ADB has a right angle at D, but triangle DCA has a right angle at A.

Therefore, the correct answer is B) Both triangles share the same side AD and congruent hypotenuses.