A kite A B C D. Point A is the top of the kite; point C is the bottom of the kite. Line A B is congruent to A D. Line B C is congruent to D C. There is a straight, vertical line drawn from point A to point C bisecting the kite.

Question

Leah wants to explain why △ABC≅△ADC
. She plans to use the SAS Congruence Theorem. She was told that AB¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯
and that AC¯¯¯¯¯¯¯¯
is the angle bisector of ∠BAD
. What sides and angles will Leah say are congruent in her explanation of why △ABC≅△ADC
using the SAS Congruence Theorem?

Option #1: AB¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯
, ∠ABC≅∠ADC
, BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯


Option #2: AB¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯
, ∠BAC≅∠DAC
, BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯


Option #3: AB¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯
, ∠BAC≅∠DAC
, AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯


Option #4: AB¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯
, ∠ABC≅∠ADC
, AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯


(1 point)
Leah will say that the sides and angles listed in Option #
are congruent in her explanation of why △ABC≅△ADC
using the SAS Congruence Theorem.

Answer 1

To use the SAS (Side-Angle-Side) Congruence Theorem to prove that triangle \( \triangle ABC \) is congruent to triangle \( \triangle ADC \), Leah needs to establish the following:

1. Side: \( AB \cong AD \) (This is given.)
2. Included Angle: \( \angle BAC \cong \angle DAC \) (This is true because line \( AC \) is the angle bisector of \( \angle BAD \).)
3. Side: \( BC \cong DC \) (This follows from the properties of the kite, as the two triangles share side \( AC \), which is congruent to itself.)

With these details in mind, the best option is:

Option #2: \( AB \cong AD \), \( \angle BAC \cong \angle DAC \), \( BC \cong DC \)

This option correctly lists the sides and angles necessary to apply the SAS Congruence Theorem to show \( \triangle ABC \cong \triangle ADC \).