Comprehensive ACT Geometry Review: Plane, Coordinate, and Trigonometry

Vertical Angles

Angles opposite each other when two lines intersect; always congruent.

Supplementary Angles

Two angles that add up to 180°.

Complementary Angles

Two angles that add up to 90°.

Transversal

A line that cuts across two parallel lines.

Alternate Interior Angles

Equal angles formed when a transversal crosses parallel lines.

Corresponding Angles

Equal angles in matching corners when a transversal crosses parallel lines.

Consecutive Interior Angles

Angles that sum to 180° formed by a transversal crossing parallel lines.

Interior Angle Sum Formula

For any polygon with n sides, the sum is (n-2) × 180°.

Exterior Angles of a Polygon

The sum is always 360° for any convex polygon.

Congruent Figures

Figures that have the exact same shape and size.

Similarity in Geometry

Figures that have the same shape but different sizes.

AA Criterion

Two triangles are similar if they share two angle measures.

Ratio of Areas in Similar Figures

If the ratio of side lengths is k, the ratio of areas is k².

Triangle Sum Theorem

The sum of interior angles in any triangle is always 180°.

Triangle Inequality Theorem

The sum of any two sides must be greater than the length of the third side.

Pythagorean Theorem

For right triangles, a² + b² = c².

Common Pythagorean Triples

3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Special Right Triangle Ratios

45-45-90: x : x : x√2; 30-60-90: x : x√3 : 2x.

SOH CAH TOA

Mnemonic for sine, cosine, and tangent ratios.

Sine Function

sin(θ) = Opposite/Hypotenuse.

Cosine Function

cos(θ) = Adjacent/Hypotenuse.

Tangent Function

tan(θ) = Opposite/Adjacent.

Circumference of a Circle

C = 2πr or C = πd.

Area of a Circle

A = πr².

Arc Length Formula

Arc Length = (n/360) × 2πr.

Sector Area Formula

Sector Area = (n/360) × πr².

Circle Equation

Standard form: (x - h)² + (y - k)² = r².

Slope-Intercept Form

y = mx + b, where m is the slope.

Distance Formula

d = √((x2 - x1)² + (y2 - y1)²).

Midpoint Formula

Midpoint = ((x1 + x2)/2, (y1 + y2)/2).