6.3-Criteria for Similarity of Triangles

6.3-Criteria for Similarity of Triangles Important Formulae

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Grade 10 → Math → Triangles → 6.3-Criteria for Similarity of Triangles

After successful completion of this topic, you should be able to:

  • Apply various criterions of similarity in order to prove whether given triangles are similar or not.
  • Show similarity of triangles in order to solve for given real life word problems.

6.3 - Criteria for Similarity of Triangles

In this subtopic, we discuss the criteria that determine whether two triangles are similar or not. Triangles are said to be similar if they have the same shape, but not necessarily the same size. There are three main criteria for the similarity of triangles in the CBSE Grade 10 Mathematics syllabus:

1. AA Criterion (Angle-Angle Criterion)

According to the AA Criterion, two triangles are similar if two corresponding angles of one triangle are equal to the two corresponding angles of another triangle. When two triangles satisfy this condition, the third angles must also be equal because the sum of the angles in any triangle is always $180^\circ$. Therefore, this criterion involves the condition:

If $ \angle A = \angle D$ and $ \angle B = \angle E$, then $ \triangle ABC \sim \triangle DEF$.

2. SSS Criterion (Side-Side-Side Criterion)

The SSS Criterion states that two triangles are similar if their corresponding sides are in the same proportion. That is, the ratio of the lengths of the corresponding sides of the two triangles must be equal. Mathematically, if the corresponding sides of two triangles are in the same ratio, then the triangles are similar. The condition is:

If $ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$, then $ \triangle ABC \sim \triangle DEF$.

3. SAS Criterion (Side-Angle-Side Criterion)

The SAS Criterion states that if in two triangles, one pair of corresponding sides is in the same ratio, and the included angle between these sides is equal, then the two triangles are similar. In other words, if the ratio of two sides of one triangle to the corresponding sides of another triangle is equal, and the angle between these sides is equal, then the two triangles will be similar. The condition is:

If $ \frac{AB}{DE} = \frac{BC}{EF}$ and $ \angle ABC = \angle DEF$, then $ \triangle ABC \sim \triangle DEF$.

Proportionality Theorem

When two triangles are similar, the corresponding sides of the triangles are proportional. This proportionality leads to various geometric properties. For example, if two triangles $ \triangle ABC$ and $ \triangle DEF$ are similar, then the following proportional relationships hold:

$ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$.

Also, the areas of two similar triangles are in the ratio of the squares of their corresponding sides. If $ \triangle ABC \sim \triangle DEF$, then:

Area of $ \triangle ABC : $ Area of $ \triangle DEF = \left( \frac{AB}{DE} \right)^2$.

Applications of Similar Triangles

Understanding similarity of triangles has many applications in real-life geometry, such as in construction, engineering, and design. One common application is finding distances that are difficult to measure directly, using the concept of similar triangles and proportionality. For example, in the case of shadows and heights, the principle of similar triangles can be used to calculate the height of an object based on the length of its shadow and the height of a reference object.

Key Points to Remember
  • Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
  • There are three main criteria for similarity: AA, SSS, and SAS.
  • The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides.

6.3- Criteria for Similarity of Triangles

त्रिकोणों की समानता के लिए निम्नलिखित तीन प्रमुख मानदंड (Criteria) होते हैं:

1. AA समकक्षता (AA Similarity Criterion):

यदि दो त्रिकोणों के दो कोण समान हों, तो उनके बीच समानता स्थापित की जा सकती है। इसका मतलब यह है कि यदि किसी त्रिकोण के दो कोण दूसरे त्रिकोण के दो कोणों के बराबर हैं, तो उन दोनों त्रिकोणों के आकार समान होंगे।

वहां यह प्रक्रिया इस प्रकार दिखाई जाती है:

यदि $ \angle A = \angle P $ और $ \angle B = \angle Q $, तो $ \triangle ABC \sim \triangle PQR $।

2. SSS समकक्षता (SSS Similarity Criterion):

यदि दो त्रिकोणों के समकक्ष (corresponding) पक्षों की लंबाई अनुपात में समान हो, तो वे त्रिकोण समान होते हैं। इसे SSS समानता सिद्धांत कहते हैं।

इसमें, यदि $ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{PR} $, तो $ \triangle ABC \sim \triangle PQR $।

3. SAS समकक्षता (SAS Similarity Criterion):

यदि दो त्रिकोणों के एक कोण समान हो और उनके संबंधित दो पक्षों का अनुपात समान हो, तो उन त्रिकोणों के बीच समानता स्थापित की जा सकती है। इसे SAS समानता सिद्धांत कहते हैं।

इसके लिए, यदि $ \angle A = \angle P $ और $ \frac{AB}{PQ} = \frac{AC}{PR} $, तो $ \triangle ABC \sim \triangle PQR $।

इन तीनों मानदंडों को लागू करके हम यह सुनिश्चित कर सकते हैं कि दो त्रिकोण समान हैं। इनका उपयोग त्रिकोणों के आकार और अनुपात के बारे में जानकारी प्राप्त करने के लिए किया जाता है।

In Fig. 6.37, if $\Delta$ABE $\cong$ $\Delta$ACD, show that $\Delta$ADE $\sim$ $\Delta$ABC.

Solution:
Since $\Delta$ABE $\cong$ $\Delta$ACD:

Side AE = Side AD (CPCT)
Side AB = Side AC (CPCT)

Taking the ratio:
$\dfrac{AD}{AB} = \dfrac{AE}{AC}$
Also, $\angle$ A is common angle.

Using SAS criterion,
$\Delta$ADE $\sim$ $\Delta$ABC.

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