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**Question 1. ABCD is a square. P is any point inside it, such that Î”PQR is another square. Prove that AP = CR**

**Hint: **Join AP and CR.

In Î”ADP and Î”CDR, we have :

AD = CD [sides of a square]

âˆ ADP = âˆ CDR [each = 90Â° - âˆ PDC]

DP = DR [side of a square]

â‡’ Î”ADP â‰Œ Î”CDR [SAS congruence]

â‡’ AP = CR [C.P.C.T]

**Question 2. E and F are the mid points of sides AB, AC of **Î”**ABC. CE and BF are produced to X and Y respectively, such that EX = CE and FY = BF. AX and AY are joined. Find in your figure, a triangle congruent to **Î”**AEX and demonstrate the congruency. Show that XAY is a st. line.**

**Hint: **Prove Î”AEX â‰Œ Î”BEC [By SAS congruency]

â‡’ âˆ XAE = âˆ CBE [c.p.c.t.]

â‡’ âˆ XAB = âˆ CBA

But they form a pair of co-interior angles.

â‡’ XA || BC ...(1)

Similarly, Î”AFY â‰Œ Î”CFB

â‡’ AY || BC ...(2)

from (1) and (2) XAY is a st. line.

**Question 3. In the adjacent figure, BA || DF and CA || EG. If BD = EC then prove that BG = DF and EG = CF.**

**Hint:** In Î”GBE and Î”FDC âˆ ABC = âˆ FDE and âˆ DED = âˆ ACB

also BE = DC

âˆ´ Î”GBE â‰Œ Î”FDC [ASA congruency]

â‡’ BG = DF and EG = CF

**Question 4. ABCD is a square. M is the mid point of AB and PQ âŠ¥ CM meets AD at P. CB produced meet at Q. Prove that (i) PA = BQ and (ii) CP = AB + PA**

**Hint:** Prove, Î”AMP â‰Œ Î”BMQ [ASA cong.]

â‡’ MP = MQ and PA = QB [c.p.c.t.]

â‡’ PA = BQ

Again, prove, âˆ†CMP â‰Œ Î”CMQ [SAS cong.]

â‡’ CP = CQ [c.p.c.t.]

â‡’ CP = CB + BQ = AB + PA

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