Question

given: \\(\angle tsr\\) and \\(\angle qrs\\) are right angles; \\(\angle t \cong \angle q\\)
prove: \\(\delta tsr \cong \delta qrs\\)

step 1: we know that \\(\angle tsr \cong \angle qrs\\) because all right angles are congruent.
step 2: we know that \\(\angle t \cong \angle q\\) because it is given.
step 3: we know that \\(\overline{sr} \cong \overline{rs}\\) because of the reflexive property.
step 4: \\(\delta tsr \cong \delta qrs\\) because

of the asa congruence theorem.
of the aas congruence theorem.
of the third angle theorem.
all right triangles are congruent.

Explanation:

Identify the given congruent parts

We are given:

• \(\angle TSR \cong \angle QRS\) (right angles, Step 1)
• \(\angle T \cong \angle Q\) (given, Step 2)

Identify the shared side

Using the Reflexive Property of Congruence:

• \(\overline{SR} \cong \overline{RS}\) (Step 3)

Determine the congruence theorem

We have two pairs of congruent angles and a non-included congruent side:

• Angle: \(\angle T \cong \angle Q\)
• Angle: \(\angle TSR \cong \angle QRS\)
• Side: \(\overline{SR} \cong \overline{RS}\)

Using the AAS Congruence Theorem, the triangles are congruent.

Answer:

• of the ASA congruence theorem.
• of the AAS congruence theorem. (Correct answer)
• of the third angle theorem.
• all right triangles are congruent.