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Triangles
EXERCISE 6.1
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
State whether the following quadrilaterals are similar or not: Fig. 6.8 79 6.3 Similarity of What can you say about the similarity of two ? You may recall that triangle is also a polygon. So, we can state the same conditions for the similarity of two . That is: Two are similiar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion). Note that if corresponding angles of two are equal, then they are known as equiangular . A famous Greek mathematician Thales gave an important truth relating to two equiangular which is as follows: The ratio of any two corresponding sides in two equiangular is always the same. It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same. To understand the Basic Proportionality
Step-by-Step Solution
Key Concept: Two polygons (including quadrilaterals) are similar if (i) their corresponding angles are equal (equi‑angular) and (ii) the lengths of their corresponding sides are in the same ratio. For quadrilaterals, we must verify both the angle‑equality and the side‑proportionality conditions.
1. Identify the two quadrilaterals – In Fig. 6.8 the quadrilaterals are labelled $ABCD$ and $A'B'C'D'$.
2. Compare the corresponding angles –
- Measure/observe $\angle A$, $\angle B$, $\angle C$, $\angle D$ of the first quadrilateral.
- Measure/observe $\angle A'$, $\angle B'$, $\angle C'$, $\angle D'$ of the second quadrilateral.
- From the figure we see that $\angle A
eq \angle A'$, $\angle B
eq \angle B'$ etc.; hence the two quadrilaterals are not equi‑angular.
3. Check the side ratios (optional) – Even if the angles were equal, we would need to verify that
$$\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CD}{C'D'} = \frac{DA}{D'A'}.$$
Using the given lengths (or the lack of a constant ratio) we find that the ratios are not all equal.
4. Conclusion – Since the necessary condition of equal corresponding angles fails (and the side‑ratio condition also fails), the quadrilaterals are not similar.
5. Answer – The quadrilaterals shown in Fig. 6.8 are not similar.
2. Compare the corresponding angles –
- Measure/observe $\angle A$, $\angle B$, $\angle C$, $\angle D$ of the first quadrilateral.
- Measure/observe $\angle A'$, $\angle B'$, $\angle C'$, $\angle D'$ of the second quadrilateral.
- From the figure we see that $\angle A
eq \angle A'$, $\angle B
eq \angle B'$ etc.; hence the two quadrilaterals are not equi‑angular.
3. Check the side ratios (optional) – Even if the angles were equal, we would need to verify that
$$\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CD}{C'D'} = \frac{DA}{D'A'}.$$
Using the given lengths (or the lack of a constant ratio) we find that the ratios are not all equal.
4. Conclusion – Since the necessary condition of equal corresponding angles fails (and the side‑ratio condition also fails), the quadrilaterals are not similar.
5. Answer – The quadrilaterals shown in Fig. 6.8 are not similar.
Correct Answer: The two quadrilaterals are not similar because their corresponding angles are not equal (and consequently the side ratios are not constant).
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