Question

name:

1. complete the proof. given: ∠a and ∠d are right angles, \\(\overline{ab} \cong \overline{de}\\) prove: \\(\triangle abe \cong \triangle deb\\)

table with statements and reasons columns:

1. ∠a and ∠d are right angles, \\(\overline{ab} \cong \overline{de}\\) | 1) given
2. | 2) definition of right triangles
3. | 3) reflexive property of congruence
4. \\(\triangle abe \cong \triangle deb\\) | 4)

image of a rectangle abe d with right angles at a and d, and segments ab and de marked congruent, and diagonal be drawn

Explanation:

Step1: Identify Right Triangles

From the given right angles ($\angle A$ and $\angle D$), by the definition of right triangles, $\triangle ABE$ and $\triangle DEB$ are right - angled at $A$ and $D$ respectively. So the statement for step 2 is $\triangle ABE$ and $\triangle DEB$ are right triangles.

Step2: Apply Reflexive Property

The reflexive property of congruence states that a segment is congruent to itself. So for the common side $\overline{BE}$, we have $\overline{BE} \cong \overline{BE}$, which is the statement for step 3.

Step3: Prove Triangle Congruence

We know that $\triangle ABE$ and $\triangle DEB$ are right triangles (step 2), $\overline{AB} \cong \overline{DE}$ (given), and $\overline{BE} \cong \overline{BE}$ (step 3). By the Hypotenuse - Leg (HL) Congruence Theorem, which states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent, we can conclude that $\triangle ABE \cong \triangle DEB$. So the reason for step 4 is the HL Congruence Theorem.

Answer:

1. $\triangle ABE$ and $\triangle DEB$ are right triangles
2. $\overline{BE} \cong \overline{BE}$
3. HL (Hypotenuse - Leg) Congruence Theorem