Question

In the given figure, $\angle YXZ=\angle XPZ,$ then, $\frac{ZX}{ZY}$ is equal to:

• A. $ZY\times ZP$
• B. $X{Z}^2$
• C. $\frac{PZ}{XZ}$
• D. $P{Z}^2$

Answer 1

Answer: (C)

$\frac{PZ}{XZ}$

AAA postulate for symmetry :

If all three corresponding interior angles of 2 triangles are the same, then the two triangles are similar.

In general, if two corresponding angles are equal, then the remaining angle will also be equal, as they always add up to ${180}^o$.

In figure (i), if $\angle A=\angle D,\angle B=\angle E,\angle C=\angle F$ by AAA postulate we can say that 2 triangles are similar.

In fig (2) given $\angle YXZ=\angle XPZ$

We can divide the given figure (iii) into two separate triangles.

Consider $△XYZ\text{and}△PXZ$.

$\angle YXZ=\angle XPZ$ ....(given)

$\angle XZY=\angle XZP$ ...($\because$ they are same )

$\angle XYZ=\angle PXZ$ ...($\because$ sum of angles = 180°)

Hence, by AAA postulate,

$△XYZ\sim$$△PXZ$

$\therefore \frac{ZX}{ZY}=\frac{PZ}{XZ}$ ....cpst