Chapter Notes: Triangles

# Triangles Chapter Notes - Mathematics (Maths) Class 10

### Similarity

• Two geometrical figures are said to be similar figures, if they have same shape but not necessarily the same size.
Or
A shape is said to be similar to other, if the ratio of their corresponding sides is equal and the corresponding angles are equal.
• Two polygons having the same number of sides are similar, if:
(i) all the corresponding angles are equal and
(ii) all the corresponding sides are in the same ratio or proportion.
If only one condition from (i) and (ii) is true for two polygons, then they cannot be similar.
• The ratio that compares the measurements of two similar shapes, is called the scale factor or representative fraction. It is equal to the ratio of corresponding sides of two figures.
We can use the ratio of corresponding sides to find unknown sides of similar shapes.
• Two triangles are said to be similar triangles, if their corresponding angles are equal and their corresponding sides are proportional (i.e., the ratios between the lengths of corresponding sides are equal).
For example:
If in ∆ABC and ∆PQR
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and,
AB/PQ = BC/QR = AC/PR
The, △ABC∼△PQR
where, symbol ∼ is read as, ‘is similar to’.
Conversely
If △ABC is similar to △PQR, then
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and,
AB/PQ = BC/QR = AC/PR

Note: The ratio of any two corresponding sides in two equiangular triangles is always the same.

• If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then other two sides are divided in the same ratio.
This theorem is known as Basic Proportionality Theorem (BPT) or Thales theorem.
• If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side (converse of basic proportionality theorem).
• Let us consider two parallel lines l and m and draw one transversal line t, which intersect l and m at P and Q. Then,

Corresponding Angles
The angles on the same side of a transversal line are known as the corresponding angles, if both lie either above the two lines or below the two lines. i.e.,
(i) ∠1and ∠5
(ii) ∠2 and ∠6
(iii) ∠4 and ∠8
(iv) ∠3 and ∠7

Alternate Interior Angles
The following pairs of angles are called pairs of alternate interior angles
(i) ∠3 and ∠5
(ii) ∠2and ∠8

Consecutive Interior Angles
The pairs of interior angles on same side of the transversal line are called pair of consecutive interior angles.
(i) ∠2 and ∠5
(ii) ∠3 and ∠8

Alternate Exterior Angles
The following pair of angles are called alternate exterior angles
(i) ∠1and ∠7
(ii) ∠4and∠6

Vertically Opposite Angles
The following pair of angles are called vertically opposite angles.
(i) ∠1and ∠3
(ii) ∠2and ∠4
(iii) ∠5 and ∠7
(iv) ∠6and ∠8

• The line segment joining the mid-points of any two sides of a triangle is parallel to the third side. (Mid-point theorem)
• The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. (Angle bisector theorem)

AAA Similarity

• (In two triangles, if corresponding angles are equal, then their corresponding sides are in the same ratio i.e., they are proportional and hence the two triangles are similar.
• If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar (because by the angle sum property of a triangle, their third angle will also be equal) and it is called AAA similarity.
The document Triangles Chapter Notes | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Triangles Chapter Notes - Mathematics (Maths) Class 10

 1. What are the different types of triangles?
Ans. There are several types of triangles, including equilateral triangles (all sides and angles are equal), isosceles triangles (two sides and two angles are equal), scalene triangles (no sides or angles are equal), and right triangles (one angle is a right angle).
 2. How do you find the area of a triangle?
Ans. The area of a triangle can be found using the formula A = (1/2) * base * height, where the base is the length of the triangle's base and the height is the perpendicular distance from the base to the opposite vertex.
 3. What is the Pythagorean theorem?
Ans. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It can be written as a^2 + b^2 = c^2, where c represents the length of the hypotenuse.
 4. How do you determine if three side lengths form a triangle?
Ans. To determine if three side lengths form a triangle, use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is satisfied for all three combinations of side lengths, then a triangle can be formed.
 5. How can you prove two triangles are congruent?
Ans. Two triangles can be proven congruent if their corresponding sides and angles are congruent. There are several methods to prove congruence, including the Side-Side-Side (SSS) criterion, the Side-Angle-Side (SAS) criterion, the Angle-Side-Angle (ASA) criterion, and the Hypotenuse-Leg (HL) criterion for right triangles.

## Mathematics (Maths) Class 10

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