How do I start a geometry proof?

Write down what is given and what is to be proved, then look for the theorem whose conclusion matches the goal. Work backwards from that theorem's requirements.

Is SSA a valid congruence rule?

No. Two sides and a non-included angle can describe two different triangles. The ambiguous case. It only works when the angle is right, which is the RHS rule.

When do I use radians for arcs?

Whenever you use the compact formulas for arc length and sector area. Those forms assume radians; with degrees you must multiply by the fraction of the full turn instead.

Do I need to memorise every volume formula?

No. Memorise base area times height, and remember that cones, pyramids and spheres carry the extra fractional factors. That reduces a dozen formulas to three ideas.

Geometry study guide

Geometry trains deduction. Each result follows from earlier results, so the fastest way to improve is to learn which theorem licenses which conclusion rather than memorising diagrams.

Practise trigonometry

Angles and parallel lines

A transversal crossing parallel lines creates equal corresponding angles, equal alternate angles, and supplementary co-interior angles. These three facts unlock most introductory angle-chasing problems.

Angle sums are your consistency check: 180 degrees in a triangle, 360 around a point, and (n−2)·180 inside any polygon. If your chase produces a contradiction, one assumed parallel is wrong.

triangle = 18 0^{\circ}

(n - 2) 18 0^{\circ}

Congruence and similarity

Congruent triangles are identical in size and shape, established by SSS, SAS, ASA, AAS or RHS. Similar triangles share shape only, established by AA, SAS ratio, or SSS ratio.

Similarity is the more useful tool in practice because it gives proportions. Once two triangles are similar, corresponding sides are in a fixed ratio and unknown lengths follow from a single proportion.

Right triangles and circles

The Pythagorean theorem and the special ratios of 30-60-90 and 45-45-90 triangles cover a large share of exam questions without any trigonometry.

Circle theorems all express the same underlying symmetry: an inscribed angle is half the central angle on the same arc, angles in a semicircle are right angles, and a tangent meets a radius at right angles.

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

A = π r^{2}

θ_{insc} = \frac{1}{2} θ_{cent}

Area, volume and coordinates

Every area formula reduces to base times height, adjusted by a factor for triangles and trapezoids. Every prism volume is base area times height, and every tapering solid carries an extra factor of one third.

Coordinate geometry turns diagrams into algebra: distance is the Pythagorean theorem, midpoints are averages, and slope conditions detect parallel and perpendicular sides in proofs.

V = B h

V = \frac{1}{3} B h

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Frequently asked questions

How do I start a geometry proof?: Write down what is given and what is to be proved, then look for the theorem whose conclusion matches the goal. Work backwards from that theorem's requirements.
Is SSA a valid congruence rule?: No. Two sides and a non-included angle can describe two different triangles. The ambiguous case. It only works when the angle is right, which is the RHS rule.
When do I use radians for arcs?: Whenever you use the compact formulas for arc length and sector area. Those forms assume radians; with degrees you must multiply by the fraction of the full turn instead.
Do I need to memorise every volume formula?: No. Memorise base area times height, and remember that cones, pyramids and spheres carry the extra fractional factors. That reduces a dozen formulas to three ideas.

Cheat sheets