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Chapter 5: Q8 (page 182)

Given that 鈻�PQRS, where PX炉bisects 鈭�QPR, RY炉bisects 鈭�SRP. Then prove thatRYPXis a parallelogram.

Short Answer

Expert verified

Therefore, RYPX is a parallelogram.

Step by step solution

01

Step 1. Check the figure.

Consider the figure.

02

Step 2. Step description.

Consider that PQRS is a parallelogram. Thus, the line PS is parallel to the line RQ.

Hence the line PY is parallel to the line RX.

By the alternate interior angle theorem, it is clear that 鈭�QPR=鈭�SRP 鈥�... (1)

03

Step 3. Step description.

Consider the angles 鈭�QPRand 鈭�SRP. Thus, from the figure

鈭�SRP=2鈭�YRP

鈭�QPR=2鈭�XPR

Use the equation (1) as follows:

2鈭�XPR=2鈭�YRP鈭�XPR=鈭�YRP

Now, the angles 鈭�XPRand 鈭�YRPare the alternate interior angles for the line RY,PX.

Thus, the lines RY,PXare parallel.

Therefore, RYPX is a parallelogram.

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