Class 7 — Mathematics (NCERT)

The Triangle and its Properties

Class 7

✓By the end of this lesson students will be able to state and apply the angle sum property of a triangle.
✓By the end of this lesson students will be able to state and apply the exterior angle property of a triangle.
✓By the end of this lesson students will be able to state and apply the Pythagoras property for right-angled triangles.
✓By the end of this lesson students will be able to solve problems involving these properties to find unknown angles or side lengths.

Key concepts

Angle Sum Property of a Triangle

The sum of the measures of the three interior angles of any triangle is always 180 degrees. This property holds true for all types of triangles, whether acute, obtuse, or right-angled.

∠A + ∠B + ∠C = 180° (for a triangle ABC)

Exterior Angle Property of a Triangle

When a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. An exterior angle and its adjacent interior angle form a linear pair, meaning their sum is 180°.

Exterior Angle = Sum of the two interior opposite angles

Pythagoras Property (for Right-Angled Triangles)

In a right-angled 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 (legs). This property is only applicable to right-angled triangles.

Hypotenuse² = Base² + Perpendicular² (or c² = a² + b²)

Key facts to remember

1The sum of the three interior angles of any triangle is always 180°.
2An exterior angle of a triangle is equal to the sum of its two interior opposite angles.
3The sum of an exterior angle and its adjacent interior angle is 180° (linear pair).
4The Pythagoras property applies only to right-angled triangles.
5In a right-angled triangle, the side opposite the right angle is called the hypotenuse.
6Pythagoras Property: Hypotenuse² = Base² + Perpendicular².

Worked examples

Example 1

In ΔPQR, if ∠P = 60° and ∠Q = 70°, find the measure of ∠R.

IWe know that the sum of the angles in a triangle is 180° (Angle Sum Property).

IISo, ∠P + ∠Q + ∠R = 180°

IIISubstitute the given values: 60° + 70° + ∠R = 180°

IVSimplify: 130° + ∠R = 180°

VSubtract 130° from both sides: ∠R = 180° - 130°

VI∠R = 50°

Answer

The measure of ∠R is 50°.

Always state the property you are using in your solution.

Example 2

In the given figure, side QR of ΔPQR is produced to a point S. If ∠P = 50° and ∠Q = 60°, find the measure of the exterior angle ∠PRS.

IAccording to the Exterior Angle Property of a triangle, the exterior angle ∠PRS is equal to the sum of the two interior opposite angles.

IISo, ∠PRS = ∠P + ∠Q

IIISubstitute the given values: ∠PRS = 50° + 60°

IVSimplify: ∠PRS = 110°

Answer

The measure of the exterior angle ∠PRS is 110°.

Alternatively, you could first find ∠PRQ using the Angle Sum Property, then use the linear pair property (∠PRQ + ∠PRS = 180°) to find ∠PRS.

Example 3

A right-angled triangle has sides of length 3 cm and 4 cm forming the right angle. Find the length of its hypotenuse.

ILet the two sides forming the right angle be 'a' and 'b', and the hypotenuse be 'c'.

IIGiven: a = 3 cm, b = 4 cm.

IIIAccording to the Pythagoras Property: c² = a² + b²

IVSubstitute the values: c² = 3² + 4²

VCalculate the squares: c² = 9 + 16

VIAdd the values: c² = 25

VIITake the square root of both sides: c = √25

VIIIc = 5 cm

Answer

The length of the hypotenuse is 5 cm.

Remember that the hypotenuse is always the longest side in a right-angled triangle.

Common mistakes

✗Applying the Pythagoras property to triangles that are not right-angled.
✗Incorrectly identifying the hypotenuse in a right-angled triangle (it's always opposite the right angle).
✗Confusing interior and exterior angles, or using the wrong interior angles for the exterior angle property.
✗Making calculation errors when squaring numbers or finding square roots.
✗Forgetting to state the property used when solving problems, which is important for full marks.

Exam tips

★Always draw a neat diagram and label it correctly with the given information.
★Clearly state the property you are using at each step of your solution.
★Show all steps of your working, even for simple calculations.
★Double-check your calculations, especially when dealing with squares and square roots.
★Practice identifying the hypotenuse quickly in different orientations of right-angled triangles.

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