CDS Mathematics · Triangles

Angles of a Triangle

The three angles of a triangle add to 180°, an exterior angle equals the two interior angles opposite it, and equal sides face equal angles.

All Triangles notes CDS Mathematics strategy

Why this matters

Eleven PYQs, eight of them MODERATE. Most are one line of arithmetic once the right fact is named: split 180° in a ratio, use the exterior-angle rule, or recall the angle at which two bisectors meet. The bisector results (90° + A/2 and 90° − A/2) are the only ones worth memorising outright.

Concept 1 of 3: Angle sum and the exterior angle

Walk round a triangle and you turn through a full

360^\circ

; the three interior angles are what is left of three straight lines,

3\times 180^\circ - 360^\circ = 180^\circ

. An exterior angle sits on a straight line with its own interior angle, so it equals the other two angles together.

Definition

Angle sum: $A + B + C = 180^\circ$ .
Exterior angle: if $BC$ is produced to $D$ , then $\angle ACD = A + B$ .
Ratios: angles in the ratio $p : q : r$ are $\dfrac{180^\circ}{p + q + r}$ times $p$ , $q$ and $r$ .
Equal multiples: if $pA = qB = rC = k$ , write each angle as $k$ over its multiplier and add.

Angle sum and exterior angle

A + B + C = 180^\circ, \qquad \angle ACD = A + B

Worked example

In triangle

ABC

3\angle A = 4\angle B = 6\angle C

. Find

\angle B

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (II) 2020 — Elementary Mathematics · Q78Easy

Example 1 · Triangles · Angles of a Triangle

In a triangle ABC, if

2 \angle A = 3 \angle B = 6 \angle C

, then what is

\angle A + \angle C

equal to ?

Opposite angles, not the adjacent one

The exterior angle at

C

equals

A + B

, the two angles AWAY from

C

. It is

180^\circ

minus the interior angle at

C

, not the sum including it.

Some questions answer themselves

A = B - C

with

A + B + C = 180^\circ

forces

B = 90^\circ

, so

A

is acute whatever the statements say. In a data-sufficiency item, first check whether the stem alone already decides the answer.

Drill 5 more on angle sum and the exterior angle

Concept 2 of 3: Where two angle bisectors meet

Bisect two angles of a triangle and the lines meet at a point whose angle depends only on the third angle. The internal bisectors meet at the incentre, the external ones at an excentre, and a mixed pair at a point where the angle is exactly half the third angle.

Definition

For triangle $ABC$ :

Internal bisectors of $B$ and $C$ meet at $I$ with $\angle BIC = 90^\circ + \dfrac A2$ .
External bisectors of $B$ and $C$ meet at $E$ with $\angle BEC = 90^\circ - \dfrac A2$ .
Internal bisector of $B$ and external bisector of $C$ meet at an angle of $\dfrac A2$ .

Proof of the first: in triangle $BIC$ the base angles are $\dfrac B2$ and $\dfrac C2$ , so $\angle BIC = 180^\circ - \dfrac{B + C}{2} = 180^\circ - \dfrac{180^\circ - A}{2}$ .

Bisector angles

\angle BIC = 90^\circ + \tfrac{A}{2}, \qquad \angle BEC = 90^\circ - \tfrac{A}{2}

Worked example

In triangle

ABC

A = 70^\circ

. Find the angle at which (a) the internal bisectors of

B

and

C

meet, (b) the external bisectors of

B

and

C

meet.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — Elementary Mathematics · Q18Moderate

Example 2 · Triangles · Angles of a Triangle

The measure of an angle formed by the bisectors of the angles A and C of the triangle ABC is

130^\circ

. What is the measure of the angle B ?

Two angles form where the bisectors cross

Crossing lines make an angle and its supplement, say

115^\circ

and

65^\circ

. The angle

\angle BIC

faces the side

BC

and is always obtuse, so use the larger one; the smaller would make the third angle negative.

Drill 2 more on where two angle bisectors meet

Concept 3 of 3: Isosceles triangles and parallel lines

Equal sides face equal angles, so one angle of an isosceles triangle fixes the other two. Parallel lines copy angles across a transversal. Most figure questions chain these two facts.

Definition

Isosceles: $AB = AC$ gives $\angle B = \angle C$ , and conversely.
Parallels: corresponding angles are equal and alternate angles are equal.
Midpoint of the hypotenuse: it is the same distance from all three vertices, so it cuts a right triangle into two isosceles triangles.

Base angles of an isosceles triangle

AB = AC \;\Rightarrow\; \angle B = \angle C = 90^\circ - \tfrac{A}{2}

Worked example

In triangle

ABC

AB = AC

and

BC

is produced to

D

with

\angle ACD = 110^\circ

. Find

\angle BAC

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — Elementary Mathematics · Q79Moderate

Example 3 · Triangles · Angles of a Triangle

In a triangle

ABC

AB = AC

and

BC

is produced to

D

such that

\angle ACD = x

, then what is

\angle BAC

equal to?

Equal angles face the equal sides

AB = AC

makes

\angle B = \angle C

, the angles at the ends of the base, not

\angle A = \angle B

. Name the angle opposite each equal side before you write it down.

Drill 1 more on isosceles triangles and parallel lines

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 and the exterior angle
Angle sum and exterior angle
$A + B + C = 180^\circ, \qquad \angle ACD = A + B$
Where two angle bisectors meet
Bisector angles
$\angle BIC = 90^\circ + \tfrac{A}{2}, \qquad \angle BEC = 90^\circ - \tfrac{A}{2}$
Isosceles triangles and parallel lines
Base angles of an isosceles triangle
$AB = AC \;\Rightarrow\; \angle B = \angle C = 90^\circ - \tfrac{A}{2}$

Watch out for (4)

Opposite angles, not the adjacent one
→ Angle sum and the exterior angle
Some questions answer themselves
→ Angle sum and the exterior angle
Two angles form where the bisectors cross
→ Where two angle bisectors meet
Equal angles face the equal sides
→ Isosceles triangles and parallel lines

Test yourself on Triangles

20 past CDS questions from this chapter, timed at 24 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 11 questions