EQN-04-01

Angles & Triangles

The angle facts and triangle rules that solve the majority of GAT-E geometry questions.

~6 min read•Difficulty 2/5

Angles on a straight line add to 180°.
Angles around a point add to 360°.
Vertically opposite angles (formed by two crossing lines) are equal.
With parallel lines cut by a transversal: corresponding and alternate angles are equal; co-interior angles add to 180°.

The three interior angles add to 180°.
An exterior angle equals the sum of the two opposite interior angles.
Equal sides face equal angles (isosceles); all equal $\to$ equilateral (each angle 60°).

Pythagoras: for legs a, b and hypotenuse c,

a^{2} + b^{2} = c^{2}

Memorize the common integer triples: 3-4-5, 5-12-13, 8-15-17 (and their multiples like 6-8-10).

45 - 45 - 90 : sides in ratio 1 : 1 : 2

30 - 60 - 90 : sides in ratio 1 : 3 : 2

These let you find every side from just one.

Diagram

The three interior angles of any triangle add up to 180°.

Worked examples

Example 1 (difficulty 1/5)

Two angles of a triangle are 50° and 60°. What is the third angle?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A right triangle has legs 6 and 8. What is the hypotenuse?

◎ Predict the answer first:

Mark every angle you can find on the figure — one fact often unlocks the next.
Spotting a Pythagorean triple saves you from computing a square root.
A figure is "not drawn to scale" unless stated — reason from the given numbers, never by eye.

Assuming a triangle is right-angled or isosceles because it looks that way

Why: The eye trusts the drawing.

Fix: Use only stated equal sides, right-angle marks, or given measures.

Adding the legs to get the hypotenuse

(6 + 8 = 14)

Why: Confusing perimeter logic with Pythagoras.

Fix: The hypotenuse comes from

a^{2} + b^{2} = c^{2},

not

a + b .