SSC CGL Maths · Triangles

Angles of a Triangle

Every triangle's angles total 180°, an outside angle is the sum of the two inside angles away from it, and the angles opposite equal sides match.

All Triangles notes

Why this matters

Twenty-four questions in the bank, all EASY but one. Fourteen are the angle sum dressed up as a ratio, a pair of differences or a chain like 4P = 5Q = 20R. The rest are an exterior angle or an isosceles triangle, and the only skill they test is putting the equal angles at the right corners.

Concept 1 of 3: Angle sum of a triangle, with ratios and equations

Tear the three corners off a paper triangle and lay them side by side: they always make a straight line. So any two angles fix the third, and any rule that ties the angles together can be solved with that one fact.

Definition

The three angles of every triangle add to 180°.
Angles in a ratio $a : b : c$ : one part is $\frac{180^\circ}{a + b + c}$ . Multiply by the part you need. For a difference between two angles, multiply the difference of their parts.
A chain like $2A = 3B = 6C$ : set each equal to $k$ , so $A = \frac{k}{2}$ , $B = \frac{k}{3}$ , $C = \frac{k}{6}$ , add them and set the sum to 180°.
Differences like $A - B = 10^\circ$ : write every angle in terms of one of them, then use the sum.
In a right triangle the two acute angles add to 90°. If one angle is the average of the other two, it is 60°.
Kinds: acute (all angles under 90°), right (one 90°), obtuse (one over 90°). A triangle has at most one angle of 90° or more. Scalene means all angles differ, isosceles two equal, equilateral all 60°.
Fast route: for a ratio question, find one part first. The answer is almost always a whole number of degrees, which checks the arithmetic.

Angle sum

\angle A + \angle B + \angle C = 180^\circ \qquad \text{one part} = \frac{180^\circ}{a + b + c}

a, b, cthe numbers in the ratio of the angles

Worked example

The angles of a triangle are in the ratio 3 : 7 : 8. Find the largest angle and say whether the triangle is acute, right or obtuse.

Practice this conceptself-check · 5 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 23 Sep 2024, Shift 3 · Q64Easy

Example 1 · Triangles · Angles of a Triangle

\Delta PQR

, if

4\angle P = 5\angle Q = 20\angle R

, then the value of

\angle Q

is:

Dividing 180° by one part of the ratio

The size of one part is 180° divided by the sum of all the parts. Only then multiply by the part the question asks for.

Allowing two obtuse or right angles

Two angles of 90° or more already reach 180°, so a triangle has at most one right or obtuse angle.

Giving the difference in parts, not degrees

With angles in the ratio 2 : 5 : 11 the difference between the largest and smallest is 9 parts of 10°, which is 90°, not 9°.

Drill 13 more on angle sum of a triangle, with ratios and equations

Concept 2 of 3: The exterior angle of a triangle

Extend one side past a corner. The angle outside and the angle inside that corner make a straight line, and the angle inside plus the other two make 180° too. So the outside angle is exactly the other two interior angles put together.

Definition

Extend side BC beyond C to D. The exterior angle at C is $\angle ACD = 180^\circ - \angle ACB$ .
An exterior angle equals the sum of the two interior angles opposite it: $\angle ACD = \angle A + \angle B$ .
So $\angle ACD - \angle B = \angle A$ , and the exterior angle is larger than either opposite interior angle.
The three exterior angles, one at each corner, add to 360°.
Fast route: when the exterior angle and a ratio of the two opposite angles are given, split the exterior angle in that ratio directly.

Exterior angle

\angle ACD = \angle A + \angle B = 180^\circ - \angle ACB

Da point on BC extended beyond C

Worked example

In triangle PQR, side QR is extended beyond R to S. The exterior angle

\angle PRS = 128^\circ

and

\angle P = 49^\circ

. Find

\angle Q

and

\angle PRQ

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 1 · Q69Easy

Example 2 · Triangles · Angles of a Triangle

In a triangle ABC, side BC is extended to a point D. The exterior angle at C is

105^\circ

. The interior angles at A and B are in the ratio 2:5. What is the measure of the largest interior angle of the triangle?

Adding the interior angle beside the exterior one

The exterior angle at C equals ∠A + ∠B, the two angles away from C. The interior angle at C is its supplement, not part of the sum.

Answering with the exterior angle as an interior one

An exterior angle of 120° is not an angle of the triangle. The interior angle at that corner is 60°, and the largest interior angle has to be found from the three interior angles.

Drill 3 more on the exterior angle of a triangle

Concept 3 of 3: Isosceles and equilateral triangles

Fold an isosceles triangle along the line from the top corner to the middle of the base. The two halves match exactly, so the two base angles are equal.

Definition

Equal sides face equal angles. If AB = AC, then $\angle B = \angle C$ : the angles opposite the equal sides, at the two ends of the third side.
The angle between the two equal sides is the apex. Base angle $= \frac{180^\circ - \text{apex}}{2}$ , and apex $= 180^\circ - 2 \times$ base angle.
Find the apex from the letters. If AC = BC, the equal sides meet at C, so the equal angles are $\angle A$ and $\angle B$ .
Equilateral: all three sides equal and every angle 60°. A right isosceles triangle has angles 90°, 45° and 45°.
Combine with the exterior angle when a side is extended: find the base angles first.

Base angle of an isosceles triangle

\text{base angle} = \frac{180^\circ - \text{apex}}{2}

Worked example

In triangle PQR, PQ = PR and

\angle Q = 66^\circ

. Find

\angle P

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 2 · Q53Easy

Example 3 · Triangles · Angles of a Triangle

The side BC of

\triangle

ABC is produced to a point D. If AC = BC and

\angle

BAC =

70^\circ

, then find the value of

2.5\angle ACD - 1.5\angle ABC

Putting the equal angles at the wrong corners

The equal angles are opposite the equal sides. With AC = BC the equal angles are ∠A and ∠B, and ∠C is the apex.

Halving the apex to get a base angle

Each base angle is half of what is left after the apex: (180° − apex) ÷ 2. An apex of 80° gives base angles of 50°, not 40°.

Drill 5 more on isosceles and equilateral triangles

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Angle sum of a triangle, with ratios and equations
Angle sum
$\angle A + \angle B + \angle C = 180^\circ \qquad \text{one part} = \frac{180^\circ}{a + b + c}$
The exterior angle of a triangle
Exterior angle
$\angle ACD = \angle A + \angle B = 180^\circ - \angle ACB$
Isosceles and equilateral triangles
Base angle of an isosceles triangle
$\text{base angle} = \frac{180^\circ - \text{apex}}{2}$

Watch out for (7)

Dividing 180° by one part of the ratio
→ Angle sum of a triangle, with ratios and equations
Allowing two obtuse or right angles
→ Angle sum of a triangle, with ratios and equations
Giving the difference in parts, not degrees
→ Angle sum of a triangle, with ratios and equations
Adding the interior angle beside the exterior one
→ The exterior angle of a triangle
Answering with the exterior angle as an interior one
→ The exterior angle of a triangle
Putting the equal angles at the wrong corners
→ Isosceles and equilateral triangles
Halving the apex to get a base angle
→ Isosceles and equilateral triangles

Test yourself on Triangles

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Triangles chapter test Drill all 24 questions