Class 10 Maths Triangles Notes

6.1 Introduction to Triangles

A triangle is a polygon with three sides and three angles. It is one of the most fundamental shapes in geometry and plays a significant role in various mathematical principles. Triangles can be classified based on their sides or angles:

• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: Two sides are equal, and the angles opposite those sides are equal.
• Scalene Triangle: All sides and angles are different.

In this chapter, we explore a special property of triangles: similarity. Similar triangles have the same shape but not necessarily the same size. This property is extremely useful in geometry and real-world applications, such as architecture and engineering.

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6.2 Similar Figures

Before understanding similarity of triangles, it is essential to understand the concept of similar figures.

Two figures are said to be similar if:

1. They have the same shape.
2. Their corresponding angles are equal.
3. The corresponding sides are in the same ratio.

For example:

• Two circles of different sizes are similar because they have the same shape, and their corresponding angles are all 360°.
• Two rectangles of different sizes are similar if their corresponding angles are 90° and the ratio of their corresponding sides is the same.

Key Properties of Similar Figures:

• Corresponding angles are equal.
• The ratio of corresponding sides is constant (i.e., the scale factor).

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6.3 Similarity of Triangles

Two triangles are said to be similar if:

1. Their corresponding angles are equal.
2. The corresponding sides are proportional (i.e., the ratio of the lengths of corresponding sides is the same).

For instance:

• If two triangles $\triangle ABC$△ABC and $\triangle DEF$△DEF are similar, then: $$\angle A = \angle D, \, \angle B = \angle E, \, \angle C = \angle F$$∠A=∠D,∠B=∠E,∠C=∠F and $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$$DEAB​=EFBC​=FDCA​

Real-Life Example:

Consider two shadows of a tree at different times of the day. The shapes of the shadows are similar to the tree, but their sizes are different. This is an example of similar triangles, where the angles are the same, but the sides are proportional.

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6.4 Criteria for Similarity of Triangles

There are several ways to determine whether two triangles are similar. The main criteria for the similarity of triangles are:

1. AA Criterion (Angle-Angle):
   • Two triangles are similar if two of their corresponding angles are equal.
   • This criterion is enough to prove similarity because if two angles of one triangle are equal to two angles of another triangle, the third angle must also be equal (since the sum of angles in a triangle is always 180°).
   Example:
   If $\angle A = \angle D$∠A=∠D and $\angle B = \angle E$∠B=∠E, then $\triangle ABC \sim \triangle DEF$△ABC∼△DEF.
2. SSS Criterion (Side-Side-Side):
   • Two triangles are similar if the corresponding sides are in the same ratio. That is, the ratio of the lengths of corresponding sides is constant.
   • Mathematically, for two triangles $\triangle ABC$△ABC and $\triangle DEF$△DEF to be similar:
   $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$$DEAB​=EFBC​=FDCA​ Example:
   If the ratio of corresponding sides $AB : DE$AB:DE, $BC : EF$BC:EF, and $CA : FD$CA:FD is the same, then the triangles are similar.
3. SAS Criterion (Side-Angle-Side):
   • Two triangles are similar if one pair of corresponding sides are proportional, and the included angles between the corresponding sides are equal.
   • Mathematically, for two triangles $\triangle ABC$△ABC and $\triangle DEF$△DEF to be similar:
   $$\frac{AB}{DE} = \frac{BC}{EF} \quad \text{and} \quad \angle B = \angle E$$DEAB​=EFBC​and∠B=∠E This criterion checks the proportionality of the sides and equality of the included angle. Example:
   If $\frac{AB}{DE} = \frac{BC}{EF}$DEAB​=EFBC​ and $\angle B = \angle E$∠B=∠E, then $\triangle ABC \sim \triangle DEF$△ABC∼△DEF.

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6.5 Summary

In this chapter, we covered the essential concepts related to similarity of triangles:

• Similarity of Figures: Figures are similar if their corresponding angles are equal, and the corresponding sides are in the same ratio.
• Similarity of Triangles: Two triangles are similar if their corresponding angles are equal, and the corresponding sides are proportional.
• Criteria for Similarity of Triangles:
  1. AA Criterion: Two triangles are similar if two corresponding angles are equal.
  2. SSS Criterion: Two triangles are similar if the corresponding sides are in the same ratio.
  3. SAS Criterion: Two triangles are similar if one pair of corresponding sides are proportional, and the included angles are equal.

These properties and criteria are crucial in solving problems related to similarity, and they help in proving geometric theorems and solving real-world problems.

MCQs Based on the “Triangles” Chapter:

1. Which of the following is a criterion for the similarity of triangles?

a) SSS
b) SAS
c) AA
d) All of the above

Answer: d) All of the above

2. If two triangles are similar, their corresponding sides are:

a) Equal
b) Proportional
c) Different
d) None of the above

Answer: b) Proportional

3. In the AA criterion of similarity of triangles, how many angles need to be equal?

a) One
b) Two
c) Three
d) Four

Answer: b) Two

4. Which of the following is a condition for the similarity of triangles based on side-angle-side (SAS) criterion?

a) The ratio of the corresponding sides is equal.
b) The corresponding angles are equal.
c) One pair of corresponding sides is proportional, and the included angles are equal.
d) The triangles must be equilateral.

Answer: c) One pair of corresponding sides is proportional, and the included angles are equal.