Question

statements

1. ∠pqt ≅ ∠rqs

2.?

1. ∠qpt and ∠qrs are right angles.

4.?

1. δpqt ≅ δrqs

reasons
?
?
given
given
all right angles are congruent.
?

(other parts: need help?; keyboard help; and some filled/empty boxes with
(overline{pq} cong overline{rq}), asa, given, ∠qpt ≅ ∠qrs )

Explanation:

Step1: Analyze Statement 1

We need to find the reason for \( \angle PQT \cong \angle RQS \). From the diagram (or context), if \( \angle PQT \) and \( \angle RQS \) are vertical angles (or given as such, but here we see the reason for \( \angle PQT \cong \angle RQS \) can be "Vertical Angles are Congruent" or if it's given, but looking at the table, the reason for statement 1 should match the congruence of angles. Wait, actually, let's check the other statements.

Step2: Analyze Statement 2

The statement 2 is \( \overline{PQ} \cong \overline{RQ} \) (from the left column, the reason is "Given"? Wait, no, the left column has \( \overline{PQ} \cong \overline{RQ} \) with no reason yet? Wait, the table has Statements and Reasons. Let's list the statements:

1. \( \angle PQT \cong \angle RQS \) - Reason?
2. \( \overline{PQ} \cong \overline{RQ} \) - Reason? (Wait, the left column has \( \overline{PQ} \cong \overline{RQ} \) as a statement? Wait, maybe the left column is the statements and the right is reasons. Wait, the image shows:

Statements:
1. \( \angle PQT \cong \angle RQS \)
2.?
3. \( \angle QPT \) and \( \angle QRS \) are right angles.
4.?
5. \( \triangle PQT \cong \triangle RQS \)

Reasons:
1.?
2.? (with \( \overline{PQ} \cong \overline{RQ} \) in the left as a statement? Maybe the left column is the statements and the right is reasons. Let's re-express:

Statement 2: \( \overline{PQ} \cong \overline{RQ} \) (Reason: Given? Wait, the left column has "Given" for one of them. Wait, the user's image has:

Left column (Statements?):
- \( \overline{PQ} \cong \overline{RQ} \)
- (blank)
- (blank)
- Given
- \( \angle PQT \cong \angle RQS \)

Right column (Reasons?):
-?
- ASA
- Given
- All right angles are congruent.
-?

Wait, maybe it's a two - column proof. Let's correct:

Two - column proof for \( \triangle PQT \cong \triangle RQS \):

1. \( \angle PQT \cong \angle RQS \) - Reason: Vertical Angles are Congruent (or Given? But the last statement in left is \( \angle PQT \cong \angle RQS \) with reason? Wait, the user's image is a bit unclear, but let's assume the standard ASA proof.

We have:

- \( \angle PQT \cong \angle RQS \) (Statement 1) - Reason: Vertical Angles Congruence Theorem (or Given if it's given)
- \( \overline{PQ} \cong \overline{RQ} \) (Statement 2) - Reason: Given
- \( \angle QPT \) and \( \angle QRS \) are right angles (Statement 3) - Reason: Given
- \( \angle QPT \cong \angle QRS \) (Statement 4) - Reason: All right angles are congruent.
- \( \triangle PQT \cong \triangle RQS \) (Statement 5) - Reason: ASA (Angle - Side - Angle)

So let's fill in the blanks:

Statement 2: \( \overline{PQ} \cong \overline{RQ} \) - Reason: Given

Statement 4: \( \angle QPT \cong \angle QRS \) - Reason: All right angles are congruent.

Statement 1: \( \angle PQT \cong \angle RQS \) - Reason: Vertical Angles are Congruent (or if it's given, but since \( \triangle PQT \cong \triangle RQS \) by ASA, we need two angles and included side. The included side is \( \overline{PQ} \cong \overline{RQ} \), angles \( \angle PQT \cong \angle RQS \) and \( \angle QPT \cong \angle QRS \) (right angles).

So:

1. \( \angle PQT \cong \angle RQS \) - Reason: Vertical Angles are Congruent (or Given)
2. \( \overline{PQ} \cong \overline{RQ} \) - Reason: Given
3. \( \angle QPT \) and \( \angle QRS \) are right angles - Reason: Given
4. \( \angle QPT \cong \angle QRS \) - Reason: All right angles are congruent.
5. \( \triangle PQT \cong \triangle RQS \) - Reason: ASA

Now, if we need to find the reason for statement 1: If \( \angle PQT \) and \( \angle RQS \) are vertical angles, then the reason is "Vertical Angles are Congruent". For statement 2, the reason is "Given" (since \( \overline{PQ} \cong \overline{RQ} \) is a given side). For statement 4, the reason is "All right angles are congruent" as provided. For statement 5, the reason is "ASA" (Angle - Side - Angle) because we have two angles (\( \angle PQT \cong \angle RQS \), \( \angle QPT \cong \angle QRS \)) and the included side (\( \overline{PQ} \cong \overline{RQ} \)) congruent.

Answer:

1. Reason for \( \angle PQT \cong \angle RQS \): Vertical Angles are Congruent (or Given)
2. Statement 2: \( \overline{PQ} \cong \overline{RQ} \), Reason: Given
3. (Already given as Given)
4. Statement 4: \( \angle QPT \cong \angle QRS \), Reason: All right angles are congruent
5. Reason for \( \triangle PQT \cong \triangle RQS \): ASA

(If we assume the question is to fill the blanks, the main blanks are:

• Statement 2: \( \overline{PQ} \cong \overline{RQ} \) (Reason: Given)
• Statement 4: \( \angle QPT \cong \angle QRS \) (Reason: All right angles are congruent)
• Statement 1: Reason: Vertical Angles are Congruent (or Given)
• Statement 5: Reason: ASA)