Overview

Geometry basics are about recognizing standard angle and length facts quickly. Draw clear diagrams, mark equal lengths, and use symmetry.

The fastest solutions usually come from one or two core facts applied cleanly.

Key Ideas

• Triangle angle sum is $180^\circ$, and exterior angles equal the sum of two remote interior angles.
• Similar triangles provide proportional sides and equal angles.
• Areas can be computed via base-height or by decomposing into simpler shapes.
• Circles add powerful angle facts, especially inscribed angles.

Core Facts

Triangle Basics

• Sum of angles: $\angle A + \angle B + \angle C = 180^\circ$.
• Exterior angle equals sum of two remote interior angles.
• Isosceles triangles have equal base angles.

Similarity

If two triangles are similar, corresponding sides are in a constant ratio $k$. Areas scale by $k^2$.

Area

• Base-height: $A=\frac{1}{2}bh$.
• Heron: $A=\sqrt{s(s-a)(s-b)(s-c)}$.
• Trig area: $A=\frac{1}{2}ab\sin C$.

Circle Angles

• Inscribed angle is half the central angle on the same arc.
• Angle in a semicircle is $90^\circ$.

Worked Example

In a right triangle with legs $6$ and $8$, find the radius of the incircle.

The hypotenuse is $10$, so the area is $\frac{1}{2}\cdot 6\cdot 8=24$. The semiperimeter is $\frac{6+8+10}{2}=12$. The inradius is $r = \frac{\text{area}}{s} = \frac{24}{12}=2$.

More Examples

Example 1: Similarity Scale

Two similar triangles have side ratio $2:3$. If the smaller area is $8$, the larger area is $8\cdot (3/2)^2 = 18$.

Example 2: Angle Chase

If a triangle has angles $x$, $2x$, and $3x$, then $6x=180^\circ$ so $x=30^\circ$.

Example 3: Circle Angle

An inscribed angle subtends a $100^\circ$ arc. Its measure is $50^\circ$.

Strategy Checklist

• Mark equal lengths and right angles.
• Look for similar triangles early.
• Use area ratios when length ratios are known.
• In circle problems, check for inscribed or central angle relations.

Common Pitfalls

• Forgetting to label right angles or congruent segments.
• Mixing up circumradius and inradius formulas.
• Assuming similarity without proving it.

Practice Problems