ABCD is a square. P, Q and Rare the points on AB, BC and CD respectively, such that AP = BQ = CR. Prove that:
If angle PQR is a rt. Angle find angle PRQ

Answer

To prove that if angle PQR is a right angle, then angle PRQ is 45 degrees, we will follow these steps:

### Step 1: Understand the Configuration
We have a square ABCD with points P, Q, and R on sides AB, BC, and CD respectively such that AP = BQ = CR. We need to prove that if angle PQR = 90 degrees, then angle PRQ = 45 degrees.

### Step 2: Label the Lengths
Let AP = BQ = CR = x. Since ABCD is a square, all sides are equal. Therefore:
- AB = AP + PB = x + PB
- BC = BQ + QC = x + QC
- CD = CR + RD = x + RD

Since AB = BC (as they are sides of the square), we can equate:
\[ x + PB = x + QC \]
This simplifies to:
\[ PB = QC \]

### Step 3: Establish Relationships
Now, since PB = QC, we can denote PB = QC = y. Therefore, we have:
- AB = x + y
- BC = x + y
- CD = x + RD

Since all sides of the square are equal, we can also say:
\[ y = RD \]

### Step 4: Analyze the Triangles
Now we have two triangles to consider:
1. Triangle PBQ
2. Triangle QRC

From the previous steps, we established that:
- PB = QC
- BQ = CR (given)
- Angle B = Angle C = 90 degrees (as they are angles of the square)

### Step 5: Prove Congruence
Using the Side-Angle-Side (SAS) congruence criterion:
- PB = QC (from our earlier step)
- BQ = CR (given)
- Angle B = Angle C = 90 degrees

Thus, triangle PBQ is congruent to triangle QRC.

### Step 6: Corresponding Parts of Congruent Triangles
Since the triangles are congruent, we have:
\[ PQ = QR \]

### Step 7: Analyze Triangle PQR
In triangle PQR:
- We know angle PQR = 90 degrees (given).
- Since PQ = QR, triangle PQR is an isosceles triangle.

### Step 8: Find Angle PRQ
In triangle PQR, the sum of angles is 180 degrees:
\[ \text{Angle PQR} + \text{Angle PRQ} + \text{Angle QRP} = 180 \]
Substituting the known values:
\[ 90 + \text{Angle PRQ} + \text{Angle PRQ} = 180 \]
Let angle PRQ = angle QRP = y.
Then:
\[ 90 + 2y = 180 \]
\[ 2y = 180 - 90 \]
\[ 2y = 90 \]
\[ y = 45 \]

### Conclusion
Thus, angle PRQ = 45 degrees.