Q. 234.7( 27 Votes )

# In Fig., sides QP and RQ of Δ PQR are produced to points S and T respectively. If ∠ SPR = 135° and ∠ PQT = 110°, find ∠ PRQ.

Answer :

**Method 1:**

It is given in the question that:

∠SPR = 135^{O}

And,

∠PQT = 110^{o}

Now, according to the question,

∠SPR + ∠QPR = 180^{O} (SQ is a straight line)

135^{o} + ∠QPR = 180^{O}

∠QPR = 45^{O}

And,

∠PQT + ∠PQR = 180^{O} (TR is a straight line)

110^{o} + ∠PQR = 180^{O}

∠PQR = 70^{O}

Now,

∠PQR + ∠QPR + ∠PRQ = 180^{O} (Sum of the interior angles of the triangle)

70^{o} + 45^{o} + ∠PRQ = 180^{O}

115^{O} + ∠PRQ = 180^{O}

**∠****PRQ = 65 ^{O}**

**Method 2:**

It is given in the question that:

∠SPR = 135^{O}

And,

∠PQT = 110^{o}

∠PQT + ∠PQR = 180^{O} (TR is a straight line)

110^{o} + ∠PQR = 180^{O}

∠PQR = 70^{O}

Now, we know that the exterior angle of the triangle equals the sum of interior opposite angles. Therefore,

∠SPR = ∠PQR + ∠PRQ

135^{o} = 70^{o} + ∠PRQ

**∠****PRQ = 65 ^{o}**

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