Triangles: ICSE Class 7 Complete Guide

A parent asked me recently why triangles get a full chapter in Class 7 when rectangles and circles only get small sections. My answer was a question back: 'Look around your room and count the triangles.' He came back the next day and said he had found triangles in the roof trusses, in the ironing board, in the bracket holding his TV, in his daughter's rangoli, and even in the stand of his bedside lamp.

Triangles are the strongest shape in engineering, the easiest to prove things about in geometry, and the building block of every other polygon. Break a pentagon into triangles, break a hexagon into triangles, prove a property of any polygon by first proving it for triangles. That is why Selina devotes an entire chapter to them in Class 7 and why this chapter lays the foundation for everything you will learn in Class 8, 9 and 10 geometry.

This guide walks through every idea in the Selina chapter the way a patient teacher would in a one on one session. We will look at classification, the angle sum property, the exterior angle theorem, construction basics and the classic traps ICSE examiners love to set.

A triangle is a closed figure made up of three line segments. It has three sides, three vertices (corner points), and three angles (also called interior angles). The total number of parts is always six: three sides plus three angles.

We name a triangle by its vertices, usually with capital letters. Triangle $ABC$ has vertices $A$, $B$ and $C$, sides $AB$, $BC$ and $CA$, and angles $\angle A$, $\angle B$ and $\angle C$. The symbol for triangle is $\triangle$, so we write $\triangle ABC$ to save time.

A few facts that sound obvious but are frequently asked in short answer questions:

* Every triangle has exactly three sides.
* Every triangle has exactly three vertices.
* Every triangle has exactly three interior angles.
* The sum of interior angles is always $180°$ (we will prove this shortly).
* The perimeter is the sum of the lengths of the three sides.

Remember the triangle inequality: the sum of any two sides of a triangle is always greater than the third side. If you try to draw a triangle with sides $2$, $3$ and $10$, you will fail, because $2 + 3 = 5$ is less than $10$. The longer side simply cannot be reached.

There are three types of triangles when we sort them by side length.

* Equilateral triangle: all three sides are equal. As a bonus, all three angles are equal too, and each angle measures exactly $60°$ because $180 \div 3 = 60$. This is the symmetrical star of triangles.

* Isosceles triangle: at least two sides are equal. The angles opposite to the equal sides are also equal. This equal angle property is a favourite of ICSE examiners. If you see an isosceles triangle and the two base angles are given as $70°$, you can immediately say the vertex angle is $180 - 70 - 70 = 40°$.

* Scalene triangle: all three sides are different lengths. Consequently, all three angles are also different.

A simple mnemonic: Equilateral = Equal all, Isosceles = Identical two, Scalene = Separate all. It sounds silly, but it sticks.

Now sort the same triangles by the size of their biggest angle.

* Acute triangle: all three angles are less than $90°$.
* Right triangle: exactly one angle is equal to $90°$. The side opposite the right angle is called the hypotenuse and it is always the longest side.
* Obtuse triangle: exactly one angle is greater than $90°$ but less than $180°$.

A common Selina question combines both classifications. 'Can a triangle be equilateral and right angled?' The answer is no, because all angles in an equilateral triangle are $60°$, so none can be $90°$. 'Can a triangle be isosceles and right angled?' Yes, think of a square cut along its diagonal. The two triangles are both isosceles (two sides equal) and right angled (the angle at the cut is $90°$). The other two angles are each $45°$.

These combined classification questions show up in every ICSE Class 7 school test, so make sure your child can argue yes or no with reasons.

This is the single most important property in the chapter. In any triangle, the sum of the three interior angles is always $180°$.

Selina introduces this by asking you to draw any triangle, cut off the three corners and arrange them along a straight line. You will see they form a straight angle, which is $180°$.

Worked example 1:
In a triangle, two angles are $55°$ and $65°$. Find the third angle.

Solution: Let the third angle be $x$.

The third angle is $60°$.

Worked example 2:
The angles of a triangle are in the ratio $2 : 3 : 4$. Find each angle.

Solution: Let the angles be $2x$, $3x$ and $4x$.

So the angles are $40°$, $60°$ and $80°$. Check: $40 + 60 + 80 = 180$. Correct.

Worked example 3:
One angle of a triangle is twice the second angle, and the third angle is $30°$ more than the first. Find all three angles.

Solution: Let the second angle be $x$. Then the first is $2x$ and the third is $2x + 30$.

So the angles are $60°$, $30°$ and $90°$. This is a right triangle.

Extend one side of a triangle beyond a vertex. The angle formed outside the triangle between the extended side and the adjacent side is called an exterior angle. Every triangle has three exterior angles, one at each vertex.

The exterior angle theorem states: the exterior angle of a triangle is equal to the sum of the two opposite interior angles (also called the 'remote interior angles' or the 'interior opposite angles').

In symbols, if the exterior angle at vertex $C$ is $\angle ACD$, then

This saves you one step compared to using the angle sum property. Instead of finding the interior angle at $C$ first and then computing $180° - \angle C$ for the exterior, you just add the other two.

Worked example 4:
In $\triangle ABC$, $\angle A = 50°$ and $\angle B = 60°$. Find the exterior angle at $C$.

Solution: By the exterior angle theorem,

Alternatively, using the angle sum property, $\angle C = 180° - 50° - 60° = 70°$, and then the exterior angle is $180° - 70° = 110°$. Same answer, more steps. The exterior angle theorem is faster.

ICSE loves to combine the exterior angle theorem with isosceles triangle properties. Make sure your child practises at least ten mixed questions.

We mentioned earlier that the sum of any two sides of a triangle must be greater than the third side. Let us see this in a Selina style question.

Worked example 5:
Can a triangle have sides of length $4$ cm, $6$ cm and $11$ cm?

Solution: Check if the sum of any two sides exceeds the third.

* $4 + 6 = 10$. Is $10 > 11$? No.

The condition fails, so no such triangle exists.

Another example: Can a triangle have sides $5$ cm, $7$ cm and $10$ cm?

* $5 + 7 = 12 > 10$. Yes.
* $5 + 10 = 15 > 7$. Yes.
* $7 + 10 = 17 > 5$. Yes.

All three checks pass, so yes, this triangle can exist.

A practical shortcut: you only need to check whether the sum of the two smallest sides is greater than the largest. If that passes, the other two automatically pass. This saves time in exam conditions.

The six errors I see most often in ICSE Class 7 triangle answers:

1. Wrong angle sum assumption. Students write $\angle A + \angle B + \angle C = 360°$, confusing triangle with quadrilateral. It is always $180°$ for a triangle.

2. Forgetting the isosceles base angle equality. In an isosceles triangle with two equal sides, the angles opposite those sides are equal. Many students forget this and try to solve the problem as if all three angles were unknown.

3. Misidentifying the hypotenuse. In a right triangle, the hypotenuse is the side opposite the right angle, not just any long side.

4. Confusing exterior with interior angle. Writing the exterior angle equals the sum of all three interior angles instead of just the two opposite ones.

5. Skipping the triangle inequality check. Writing 'yes, a triangle with sides $4, 5, 10$ exists' without verifying. Always check.

6. Sloppy construction. Erasing arcs after finishing the construction. Examiners want to see your arcs, because they are proof of method. Do not erase them.

* A triangle has three sides, three vertices and three angles. The angle sum is always $180°$.
* Classify by sides: equilateral (all equal), isosceles (two equal), scalene (all different).
* Classify by angles: acute (all less than $90°$), right (one equals $90°$), obtuse (one greater than $90°$).
* Exterior angle theorem: the exterior angle equals the sum of the two opposite interior angles.
* Triangle inequality: the sum of any two sides must be greater than the third side.
* In an isosceles triangle, the angles opposite the equal sides are equal.
* Construction cases in Class 7: SSS, SAS and ASA. Label your figure and never erase arcs.