SOLUTION: 1) ABCD is a parallelogram if AB = 2 AD and P is the midpoint of AB.prove that angle CPD = 90. 2) If the diagonal PR and QS of a parallelogram PQRS are equal, prove that PQRS i

Algebra ->  Parallelograms -> SOLUTION: 1) ABCD is a parallelogram if AB = 2 AD and P is the midpoint of AB.prove that angle CPD = 90. 2) If the diagonal PR and QS of a parallelogram PQRS are equal, prove that PQRS i      Log On


   

Question 151452: 1) ABCD is a parallelogram if AB = 2 AD and P is the midpoint of AB.prove that angle CPD = 90.
2) If the diagonal PR and QS of a parallelogram PQRS are equal, prove that PQRS is a rectangle.
3) PQRS is a parallelogram. PS is produced to M so that SM = SR and MR is produced to meet PQ produced to N. Prove that QN = QR.
Answer by orca(409) About Me  (Show Source):
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1) ABCD is a parallelogram if AB = 2 AD and P is the midpoint of AB.prove that angle CPD = 90.
SOLUTION:
In isosceles triangle APD,
< APD = < ADP = (180 - < A)/2 = 90 - < A/2
In isosceles triangle PCB,
< BPC = < BCP = (180 - < B)/2 = 90 - < B/2
Next express < DPC in terms of < A and < B.
< DPC = 180 - < APD - < BPC = 180 - (90 - < A/2) - (90 - < B/2) = < A/2+ < B/2
= (< A + < B)/2
Note that in any parallelogram the sum of any two adjacent angles is 180. so:
< A + < B = 180
Therefore
< DPC = (< A + < B)/2 = 180/2 = 90
ALTERNATIVE SOLUTION:
First prove that
PD bisects < D
PC bisect < C
(Let Q be the midpoint of CD. As triangle APD and QPD are congruent, so < ADP = < QDP. For the same reason < BCP = < QCP)
Next
< PDQ = < D/2
< PCQ = < C/2
Thus < PDQ + < PCQ = < D/2 + < C/2 = (< D + < C)/2 = 180/2 = 90.
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2) If the diagonal PR and QS of a parallelogram PQRS are equal, prove that PQRS is a rectangle.
SOLUTION:
We need to prove its angles are 90 degree.
First prove that triangle PQR and triangle QRS are congruent.(Reason: PQ = SR, QR = QR and PR = QS)
So < Q = < R
Next note that < Q + < R = 180.
As < Q = < R, < Q = < R = 90.
So PQRS is a rectangle.
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PQRS is a parallelogram. PS is produced to M so that SM = SR and MR is produced to meet PQ produced to N. Prove that QN = QR.
SOLUTION:
We need to prove triangle QNR is an isosceles triangle.
First show that < N = < SRM and < QRN = < M ( ? )
Next show that < SRM = < M ( ? )
Conclusion: < N = < QRN ( ? ), So triangle QNR is isosceles.