Question

In the given figure, it is given that XY is equal to XZ and X is the mid point of QR. Exterior angle of QXY is equal to exterior angle of RXZ . Prove YP is equal to PZ​

Answer 1

Answer:

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Answer 2

Solution :-

→∠QXZ =∠RXY { given that exterior angle of ∠RXZ and ∠QXY are equal. }

→ ∠QXZ -∠YXZ = ∠RXY - ∠YXZ

→ ∠QXY = ∠RXZ -------- Eqn.(1)

So, in ∆QXY and ∆RXZ we have,

→ QX = RX { given that X is the mid point of QR. }

→ ∠QXY = ∠RXZ { from Eqn.(1) }

→ XY = XZ J { given }

then,

→ ∆QXY ≅ ∆RXZ { By SAS congruence rule . }

therefore,

• ∠YQX = ∠ZRX ------- Eqn.(2)
• YQ = ZR ---------- Eqn.(3)
• When two ∆'s are congruent , their corresponding sides are equal in length and their corresponding angles are equal in measure .

now, from Eqn.(2) ,

→ ∠YQX = ∠ZRX

→ ∠PQR = ∠PRQ

hence,

→ PQ = PR { The sides opposite to equal angles of a ∆ are equal in measure. }

→ PQ - YQ = PR - ZR

using Eqn.(3) ,

→ PY = PZ (Proved.)

" ∴ PY is equal to PZ ."

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