In the given figure ABCD is a square and P, Q, R are points on AB, BC and CD respectively such that AP= BQ = CR and PQR= 90° prove that PB= QC , PQ = QR and QPR = 45°
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In the given figure ABCD is a square and P, Q, R are points on AB, BC and CD respectively such that AP= BQ = CR and PQR= 90° prove that PB= QC , PQ = QR and QPR = 45°
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Answer:
Step-by-step explanation:
To prove that PB = QC, PQ = QR, and QPR = 45°, ABCD is a square,
AP = BQ = CR, and ∠PQR = 90°.
ABCD is a square. In a square, all sides are equal, and all angles are 90°.
AP = BQ = CR, lengths of AP, BQ, and CR are equal.
statement reason
In triangle BPQ and triangle CQR.
PQ = QR (given)
BP = CQ AP = BQ square sides are
equal
∠BPQ = ∠CQR = 90° ∠PQR = 90°
BPQ and CQR are congruent. Side-Angle-Side (SAS)
PB = QC. corresponding sides are equal
PQ = QR: given in the statement
∠QPR = 45°:
Since AP = BQ and ∠PQR = 90°, we can say that triangle PQR is an isosceles right triangle, with PQ = QR and one angle of 90°.
In an isosceles right triangle, the two acute angles are equal, and each measures 45°.
Therefore, ∠QPR = 45°.
Hence, proved that PB = QC, PQ = QR, and ∠QPR = 45° based on the properties of a square and congruent triangles.