CBSE Class 7 Maths — Triangles and Properties

Chapter Overview & Weightage

Triangles and their properties is one of the most foundational chapters in Class 7 Maths. Almost every geometry question in classes 8–10 builds on concepts first introduced here. In CBSE Class 7 annual exams, this chapter typically carries 8–12 marks — a mix of 1-mark MCQs, 2-mark short answers, and one 4-mark proof or construction question.

Question Type	Typical Marks	Topics Tested
MCQ / Fill in the blank	1–2	Angle sum, exterior angle, triangle inequality
Short Answer	2–3	Finding missing angles, types of triangles
Long Answer / Proof	4	Exterior angle theorem, Pythagoras (intro)

The angle sum property and the exterior angle theorem together cover nearly 60% of the marks from this chapter. Master these two and you can score full marks on most questions.

Key Concepts You Must Know

Types of triangles by sides:

Equilateral: All three sides equal, all angles = 60°
Isosceles: Two sides equal, base angles equal
Scalene: All sides different, all angles different

Types of triangles by angles:

Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°

Angle Sum Property: The sum of all interior angles of a triangle = 180°. This is the single most used fact in geometry at Class 7–10 level.

Exterior Angle Property: An exterior angle of a triangle equals the sum of its two non-adjacent (remote) interior angles.

Triangle Inequality: The sum of any two sides of a triangle is always greater than the third side. This tells us which combinations of lengths can actually form a triangle.

Pythagoras Theorem (intro): In a right triangle, the square of the hypotenuse = sum of squares of the other two sides: $a^2 + b^2 = c^2$ . Class 7 introduces this; Class 10 proves it formally.

Medians and Altitudes: A median joins a vertex to the midpoint of the opposite side. An altitude is perpendicular from a vertex to the opposite side. Every triangle has 3 medians and 3 altitudes; they each meet at a single point (centroid and orthocentre respectively).

Important Formulas

\angle A + \angle B + \angle C = 180°

Use when: You know two angles and need to find the third.

\text{Exterior angle} = \angle A + \angle B \quad \text{(two non-adjacent interior angles)}

Use when: An exterior angle is given or asked about.

a + b > c, \quad b + c > a, \quad a + c > b

Use when: You need to check if given lengths can form a triangle.

\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2

c^2 = a^2 + b^2

Use for: Right-angled triangles only.

Solved Previous Year Questions

PYQ 1 — Finding Missing Angles (2 marks)

Q: In triangle PQR, ∠P = 65° and ∠Q = 48°. Find ∠R.

Solution: By angle sum property: $\angle P + \angle Q + \angle R = 180°$

$65° + 48° + \angle R = 180°$

$\angle R = 180° - 113° = \mathbf{67°}$

PYQ 2 — Exterior Angle Theorem (3 marks)

Q: In triangle ABC, an exterior angle at vertex C measures 115°. If ∠A = 60°, find ∠B.

Solution: By exterior angle theorem: Exterior angle = ∠A + ∠B

$115° = 60° + \angle B$

$\angle B = 55°$

Verification: ∠A + ∠B + ∠C = 60° + 55° + (180° − 115°) = 60° + 55° + 65° = 180° ✓

PYQ 3 — Triangle Inequality (2 marks)

Q: Can a triangle have sides of lengths 4 cm, 7 cm, and 12 cm?

Solution: Check all three combinations:

$4 + 7 = 11 < 12$ — this condition fails!

Since the sum of two sides (4 + 7 = 11) is less than the third side (12), no, this cannot form a triangle.

PYQ 4 — Pythagoras Theorem (2 marks)

Q: A right triangle has legs 5 cm and 12 cm. Find the hypotenuse.

Solution: $c^2 = 5^2 + 12^2 = 25 + 144 = 169$

$c = \sqrt{169} = \mathbf{13 \text{ cm}}$

Difficulty Distribution

Difficulty	% of Questions	Types
Easy	50%	Direct angle sum, exterior angle with one step
Medium	35%	Multi-step angle problems, triangle inequality checks
Hard	15%	Multi-step proofs, combining Pythagoras with angle properties

Expert Strategy

First, always draw a diagram — even a rough one. Mark all given angles and lengths. This alone prevents most errors.

When working with exterior angles, students often get confused about which angles are “non-adjacent.” A simple rule: the exterior angle is formed by extending one side of the triangle. The two angles that are NOT next to that extended side are the remote interior angles.

For Pythagoras questions, always identify the hypotenuse first — it is always opposite the right angle and is always the longest side. If you try to use a leg as the hypotenuse, your equation will be wrong.

Memorise the common Pythagorean triplets: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). These appear repeatedly in exams and save calculation time.

When the question says “an exterior angle of a triangle,” draw the full picture first. Mark the exterior angle and then identify which two interior angles are non-adjacent to it. This prevents the classic mistake of adding the wrong pair of angles.

Common Traps

Trap 1 — Exterior angle vs supplementary angle: Students often confuse “exterior angle” (= sum of two remote interior angles) with “supplement of an interior angle” (= 180° minus that interior angle). Both describe the same angle — but when asked to prove a relationship, you must use the exterior angle theorem, not the supplementary angle approach, if the question specifically asks for that theorem.

Trap 2 — Pythagoras only for right triangles: The formula $a^2 + b^2 = c^2$ works ONLY when the triangle has a 90° angle. Students sometimes apply it to all triangles. If no right angle is stated or shown, do not use Pythagoras.

Trap 3 — Triangle inequality direction: The triangle inequality says the sum of TWO sides must be greater than the THIRD. Students sometimes check only one pair. Always check all three combinations — only then can you confirm a valid triangle.

Trap 4 — Equilateral vs equiangular: All equilateral triangles are equiangular (each angle = 60°), and all equiangular triangles are equilateral. These two properties always go together — you can state either one when the other is given.