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ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees?
Verified Answer
ABCD is a square . P and Q are points on DC and BC respectively such t...
a) In ΔADP and ΔDCQ
AD = DC (Sides of a square are equal)
∠ADP = ∠DCQ = 90degree
AP = DQ (Given)
∴Using RHS congruency rule, ΔADP ΔDCQ
⇒∠DAP = CDQ or ∠1 = ∠3....(1)
b) ∠DMP = ∠1 + ∠2 (Exterior angle property)
= ∠3 + ∠2 (From (1))
= ∠ADP
= 90degree
Hence, proved.
This question is part of UPSC exam. View all Class 9 courses
Most Upvoted Answer
ABCD is a square . P and Q are points on DC and BC respectively such t...
Proof:
Let's prove that angle DMP is 90 degrees.
1. Given:
- ABCD is a square.
- P and Q are points on DC and BC respectively such that AP = DQ.
- AP and DQ intersect each other at point M.
2. To Prove:
- Angle DMP = 90 degrees.
3. Proof:
- Since ABCD is a square, all sides are equal in length and all angles are 90 degrees.
- Let's consider the point E on AD such that AE = AP.
- Therefore, we have AP = AE = DQ.
- Triangle AEP is congruent to Triangle DQM (By Side-Side-Side congruence).
- Hence, angle AEP = angle DQM.
4. Property:
- Opposite angles in a square are equal.
5. Conclusion:
- Angle DMP = angle DME + angle EMP (By angle addition postulate).
- Angle DMP = angle AEP + angle EMP (By property of opposite angles in a square).
- Angle DMP = angle AEP + angle EMP = angle DQM + angle EMP (By step 3).
- Angle DMP = angle DQM + angle EMP = 90 degrees (By property of opposite angles in a square).
- Therefore, angle DMP is 90 degrees.
Thus, we have proved that angle DMP is 90 degrees.
Let's prove that angle DMP is 90 degrees.
1. Given:
- ABCD is a square.
- P and Q are points on DC and BC respectively such that AP = DQ.
- AP and DQ intersect each other at point M.
2. To Prove:
- Angle DMP = 90 degrees.
3. Proof:
- Since ABCD is a square, all sides are equal in length and all angles are 90 degrees.
- Let's consider the point E on AD such that AE = AP.
- Therefore, we have AP = AE = DQ.
- Triangle AEP is congruent to Triangle DQM (By Side-Side-Side congruence).
- Hence, angle AEP = angle DQM.
4. Property:
- Opposite angles in a square are equal.
5. Conclusion:
- Angle DMP = angle DME + angle EMP (By angle addition postulate).
- Angle DMP = angle AEP + angle EMP (By property of opposite angles in a square).
- Angle DMP = angle AEP + angle EMP = angle DQM + angle EMP (By step 3).
- Angle DMP = angle DQM + angle EMP = 90 degrees (By property of opposite angles in a square).
- Therefore, angle DMP is 90 degrees.
Thus, we have proved that angle DMP is 90 degrees.
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ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees?
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ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees?.
ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a square . P and Q are points on DC and BC respectively such that AP =DQ and AP and DQ intersect each other at point M . Prove that angle DMP = 90 degrees?.
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