Question

In the given figure, PQ || RV || XZ and QT || UW. If UY = XY and ∠PQT = 60°, then which of the following is not true?

• A. UYX is an equilateral triangle
• B. ∠VUY = 60°
• C. ∠UYZ = 2(∠SUX + ∠PQS)
• D. ∠TSU + ∠YXW = 2(∠VUY + ∠RST)

Answer 1

The correct option is C ∠UYZ = 2(∠SUX + ∠PQS)


Given: PQ || RV || XZ, QT || UW, UY = XY, ∠PQT = 60°
∠RST = ∠PQT (Corresponding angles)
⟹ ∠RST = 60°
Now, ∠RUW = ∠RST (Corresponding angles)
⟹ ∠RUW = 60°
Now, ∠UXY = ∠RUW (Alternate interior angles)
⟹ ∠UXY = 60° …..(i)
In ΔUXY, UY = XY.
Now, angles opposite to equal sides are equal.
∴ ∠YUX = ∠UXY
⟹ ∠YUX = 60° [From (i)] …..(ii)
In ΔUXY, by Angle sum property,
∠UXY + ∠YUX + ∠UYX = 180°
⟹ ∠UYX = 180° – (60° + 60°) [From (i) and (ii)]
⟹ ∠UYX = 60° …..(iii)
Therefore, ΔUYX is an equilateral triangle.
Since RV is a straight line.
∴ ∠VUY + ∠YUR = 180° (Linear pair)
⟹ ∠VUY + (∠YUX + ∠XUR) = 180°
⟹ ∠VUY = 180° – (60° + 60°) [From (ii)]
⟹ ∠VUY = 60° …..(iv)
Now, ∠VUY + ∠RST = 60° + 60° [From (iv)]
⟹ ∠VUY + ∠RST = 120° …..(v)
Now, ∠TSU + ∠SUW = 180° (Angles on same side of transversal)
⟹ ∠TSU = 180° – 60°
⟹ ∠TSU = 120° …..(vi)
Now, ∠YXW + ∠UXY = 180° (Linear pair)
⟹ ∠YXW = 180° – 60° [From (i)]
⟹ ∠YXW = 120° …..(vii)
From (vi) and (vii), we get
∠TSU + ∠YXW = 120° + 120°
⟹ ∠TSU + ∠YXW = 240°
⟹ ∠TSU + ∠YXW = 2(∠VUY + ∠RST) [From (v)]
Now, ∠SUX + ∠PQS = 60° + 60°
= 120°
∴ 2(∠SUX + ∠PQS) = 2 × 120°
= 240°
Also, ∠UYZ = 180° – ∠UYX (Linear pair)
= 180° – 60°
= 120°
Clearly, ∠UYZ ≠ 2 (∠SOX + ∠PQS)
Thus, the only incorrect equation is ∠UYZ = 2(∠SUX + ∠PQS)
Hence, the correct answer is option (c).