ACT Math Geometry: Core Ideas, Formulas, and Problem-Solving Methods

Vertical angles

Opposite angles formed when two lines intersect; vertical angles are equal.

Linear pair

Two adjacent angles that form a straight line; their measures add to 180 degrees.

Corresponding angles

Angles in the same relative position when a transversal crosses parallel lines; they are equal.

Alternate interior angles

Angles between parallel lines on opposite sides of a transversal; they are equal.

Same-side interior angles

Interior angles on the same side of a transversal through parallel lines; they are supplementary and sum to 180 degrees.

Polygon angle formulas

For an n-gon, the interior angle sum is (n - 2)180. In a regular polygon, each interior angle is ((n - 2)180)/n, each exterior angle is 360/n, and all exterior angles together total 360 degrees.

Congruent figures

Figures with the same size and shape; one can be mapped onto the other by rigid motions. Triangle congruence is often shown by SSS, SAS, ASA, AAS, or HL.

Similar figures

Figures with the same shape but not necessarily the same size; corresponding angles are equal and corresponding sides are proportional. Triangle similarity is often shown by AA, SAS similarity, or SSS similarity.

Scale factor

The constant ratio k between corresponding side lengths of similar figures; areas scale by $k^2$ and volumes scale by $k^3$.

Pythagorean theorem

In a right triangle, $a^2 + b^2 = c^2$, where c is the hypotenuse.

45-45-90 triangle

A special right triangle with side ratio 1 : 1 : $\sqrt{2}$; if each leg is x, the hypotenuse is $x \cdot \sqrt{2}$.

30-60-90 triangle

A special right triangle with side ratio 1 : $\sqrt{3} : 2$; if the shortest side is x, the longer leg is $x \cdot \sqrt{3}$ and the hypotenuse is $2x$.

Sine

In a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Cosine

In a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Tangent ratio

In a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Central angle

An angle whose vertex is at the center of a circle; its measure equals the measure of its intercepted arc.

Inscribed angle

An angle whose vertex lies on a circle; its measure is half the measure of its intercepted arc.

Tangent line to a circle

A line that touches a circle at exactly one point and is perpendicular to the radius at the point of tangency.

Sector and arc length

A sector is a slice of a circle. Its area is $\frac{\theta}{360}\pi r^2$, and its arc length is $\frac{\theta}{360}2\pi r$.

Prism and cylinder formulas

Volume is base area times height, $V = Bh$. For a cylinder, $V = \pi r^2h$ and surface area is $2 \pi r^2 + 2 \pi rh$.

Pyramid, cone, and sphere formulas

Pyramids and cones have volume $V = \frac{1}{3}Bh$; for a cone, $V = \frac{1}{3}\pi r^2h$. A sphere has surface area $4\pi r^2$ and volume $\frac{4}{3} \pi r^3$.

Distance and midpoint formulas

Between points $(x_1, y_1)$ and $(x_2, y_2)$, distance is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, and midpoint is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Slope

The steepness of a line, $m = \frac{y_2 - y_1}{x_2 - x_1}$. Parallel lines have equal slopes, and perpendicular lines have negative reciprocal slopes when both slopes exist.

Circle equation

A circle with center $(h, k)$ and radius $r$ has equation $(x - h)^2 + (y - k)^2 = r^2$.

Conic sections

The main conics are circles, parabolas, ellipses, and hyperbolas. In standard form, a circle has matching squared terms, a parabola has only one squared variable, an ellipse has added squared terms, and a hyperbola has subtracted squared terms.