Question

given: \\( \triangle a b c \\) and \\( \triangle c d a \\) are right \\( \triangle s ; \overline{a d} \cong \overline{c b} \\)
prove: \\( \triangle a b c \cong \triangle c d a \\)

statements

1. \\( \triangle a b c \\) and \\( \triangle c d a \\) are right \\( \triangle s ; \overline{a d} \cong \overline{c b} \\) 1. given

2.

1. \\( \triangle a b c \cong \triangle c d a \\)

reasons

1. reflexive property of \\( \cong \\)

3.

a. hl
b. \\( \angle d a c \cong \angle d a c \\)
c. \\( \overline{a c} \cong \overline{a c} \\)
d. ll
e. sas

Explanation:

Step1: Identify the reflexive side

The reflexive property of congruence states that a segment is congruent to itself. In right triangles \( \triangle ABC\) and \( \triangle CDA\), the hypotenuse \(AC\) is common. So, \( \overline{AC}\cong\overline{AC}\).

Step2: Apply the right - triangle congruence criterion

We have two right triangles (\(\triangle ABC\) and \(\triangle CDA\)). We know that \( \overline{AD}\cong\overline{CB}\) (given) and \( \overline{AC}\cong\overline{AC}\) (reflexive property). For right - triangles, the Hypotenuse - Leg (HL) congruence criterion states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two right triangles are congruent. Here, \(AC\) is the hypotenuse and \(AD\) (in \(\triangle CDA\)) and \(CB\) (in \(\triangle ABC\)) are legs.

Answer:

1. c. \( \overline{AC}\cong\overline{AC}\)
2. a. HL