State whether the following quadrilaterals are similar or not:

Question

Homi · Accepted Answer

Understand the Question

Two polygons of the same number of sides are similar if and only if both conditions are satisfied:
1. Their corresponding angles are equal.
2. Their corresponding sides are in the same ratio (proportional).
Both conditions must hold simultaneously; if either condition fails, the figures are not similar.

Step 1 · Check Ratio of Corresponding Sides

Diagram 1

Ratio of the corresponding sides:

\begin{aligned} \dfrac{PQ}{AB} &= \dfrac{1.5}{3} = \dfrac{1}{2} \\[0.6em] \dfrac{QR}{BC} &= \dfrac{1.5}{3} = \dfrac{1}{2} \\[0.6em] \dfrac{RS}{CD} &= \dfrac{1.5}{3} = \dfrac{1}{2} \\[0.6em] \dfrac{SP}{DA} &= \dfrac{1.5}{3} = \dfrac{1}{2} \end{aligned}

$\dfrac{PQ}{AB} = \dfrac{QR}{BC} = \dfrac{RS}{CD} = \dfrac{SP}{DA} = \dfrac{1}{2}$

The corresponding sides are in the same ratio (proportional).

Step 2 · Check Corresponding Angles

For quadrilateral $ABCD$ (square): $\angle A = \angle B = \angle C = \angle D = 90^\circ$

For quadrilateral $PQRS$ (rhombus): The angles are not $90^\circ$ ( $\angle P$ and $\angle R$ are acute, while $\angle Q$ and $\angle S$ are obtuse).

Since the corresponding angles are not equal: $\angle P \neq \angle A, \quad \angle Q \neq \angle B, \quad \angle R \neq \angle C, \quad \angle S \neq \angle D$

Therefore, the two quadrilaterals are not similar.

Answer

The two quadrilaterals are not similar.

Common Mistakes

Assuming Proportional Sides Are Sufficient: In triangles, side proportionality guarantees similarity (SSS), but in quadrilaterals and higher polygons, both proportional sides and equal corresponding angles are required.
Confusing Rhombus and Square: Both figures have equal side lengths, but a square has $90^\circ$ angles whereas a general rhombus does not.

Question 3

More questions in Exercise 6.1