Essential Geometry Rules
The handful of rules behind most geometry questions: the 180 degree triangle, the 360 degree quadrilateral, equal angles from crossing and parallel lines, isosceles triangles and similar-triangle ratios.
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Most geometry questions in aptitude tests rest on a small set of rules. Learn these few and the majority of problems become a short calculation rather than a puzzle — no construction, no theorem-hunting, just the right fact applied.
Rule 1
The two angle totals
The angles of any triangle add to 180°.
A straight line also measures 180°.
These two facts alone solve a surprising share of problems. If a question gives you two angles of a triangle, the third is already determined.
Rule 2
Equal angles from crossing lines
Vertical angles — when two lines cross, the angles opposite each other are always equal.
Corresponding angles — when a line cuts two parallel lines, angles in the same position are equal.
These let you carry a known angle to a different part of the diagram, which is often exactly what a question requires.
Rule 3
The isosceles triangle
If a triangle has two equal sides, then the two angles opposite those sides are also equal. This holds whether or not the triangle contains a right angle.
Combined with the 180° rule, one given angle is enough to find all three.
Rule 4
Quadrilaterals and similar triangles
The angles of any quadrilateral add to 360° — square, rectangle, rhombus and parallelogram alike.
In similar triangles, corresponding sides are in a constant ratio.
The similar-triangle ratio is what lets you find an unknown length from a matching pair of sides, without needing any angle at all.
Example
An isosceles triangle
In triangle ABC, cm and angle
. Find angles B and C.
Since 
, the opposite angles are equal, so 
.
⟹ B = C = 70°
Two rules combined — equal sides give equal angles, and the total is 180° — turned one given angle into all three.
Example
A quadrilateral
A quadrilateral has angles of 90°, 120° and 85°. Find the fourth.
The four angles must total 360°.
⟹ the fourth angle is 65°
Note
Mistakes to avoid
Using 180° for a quadrilateral, or 360° for a triangle.
Assuming a triangle is isosceles because it looks that way — the equal sides must be stated or marked.
Applying the corresponding-angles rule when the lines are not parallel.
Measuring angles off the diagram instead of calculating; figures are rarely to scale.
Forgetting that the equal angles in an isosceles triangle are the ones opposite the equal sides.
Summary
- A triangle's angles total 180°, and so does a straight line.
- Vertical angles are always equal; corresponding angles are equal when the lines are parallel.
- Two equal sides in a triangle give two equal opposite angles.
- Any quadrilateral's angles total 360°.
- In similar triangles, corresponding sides keep a constant ratio.