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

Angles and Parallel Lines

Geometry problems often look “diagram-heavy,” but many of them reduce to a few reliable angle relationships. When you understand how angles behave—especially when parallel lines are involved—you can turn a picture into equations.

Basic angle facts (the building blocks)

An angle measures rotation between two rays and is usually given in degrees on the ACT.

Two quick relationships show up constantly:

• Vertical angles: When two lines cross, opposite angles are equal.
• Linear pairs: Adjacent angles forming a straight line add to 180 degrees.

These matter because you can often label one angle, then “chase” around the diagram using equalities and sums to find unknowns.

Example (vertical + linear pair):
Two lines intersect. One angle is 35 degrees. Find the other three.

• The vertical angle is also 35.
• Each adjacent linear-pair angle is 180 - 35 = 145.
• The last angle (vertical to 145) is 145.

Common pitfall: mixing up vertical angles (equal) with adjacent angles (supplementary).

Parallel lines cut by a transversal

When a line (a transversal) crosses two parallel lines, several angle pairs become equal. This is powerful because it lets you copy an angle measure from one intersection to the other.

Key relationships:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Same-side (consecutive) interior angles are supplementary (sum to 180).

Why it matters: Many ACT questions give you one angle algebraically (like 3x+10) and another that’s equal or supplementary. You set up a simple equation and solve for x.

Example (alternate interior angles):
Two parallel lines are cut by a transversal. An alternate interior angle is labeled 2x+15 and its partner is 5x-30. Find x.

Alternate interior angles are equal, so:

2x + 15 = 5x - 30

Add 30 to both sides:

2x + 45 = 5x

Subtract 2x:

45 = 3x

So:

x = 15

Common pitfall: using supplementary when the angles are actually equal (or vice versa). If the diagram indicates the angles are on the “same side” inside the parallels, that’s a strong hint they sum to 180.

Polygons and interior/exterior angle sums

For an n-gon (polygon with n sides):

• Sum of interior angles:

S = (n - 2)\cdot 180

• Each interior angle of a regular polygon (all sides and angles equal):

\text{each interior} = \frac{(n - 2)\cdot 180}{n}

• Each exterior angle of a regular polygon:

\text{each exterior} = \frac{360}{n}

A deep idea that prevents errors: exterior angles “walk you around” the polygon once, so the total turn is always 360 (for any convex polygon).

Example (regular polygon):
A regular polygon has each exterior angle 30 degrees. Find n.

30 = \frac{360}{n}

n = 12

Exam Focus

• Typical question patterns:
  • Solve for x using angle relationships from a parallel-lines diagram.
  • Find missing angles in intersecting lines using vertical/linear-pair facts.
  • Determine number of sides in a regular polygon from an interior/exterior angle.
• Common mistakes:
  • Treating corresponding/alternate interior angles as supplementary instead of equal.
  • Forgetting that same-side interior angles sum to 180.
  • Mixing up interior vs exterior angle formulas (especially for regular polygons).

Congruence and Similarity

Congruence and similarity are about comparing shapes. Congruent figures are the “same size and same shape.” Similar figures are the “same shape” but can be scaled larger or smaller.

Congruent figures (rigid motions)

Two figures are congruent if one can be moved onto the other using rigid motions (translations, rotations, reflections). Rigid motions preserve distances and angles.

For triangles, you typically prove or use congruence via criteria such as:

• SSS: three corresponding sides equal
• SAS: two sides and the included angle equal
• ASA/AAS: two angles and a side equal
• HL (right triangles): hypotenuse and one leg equal

Why it matters: Once triangles are congruent, all corresponding parts match (often remembered as CPCTC: corresponding parts of congruent triangles are congruent). That lets you transfer lengths/angles to places you don’t directly know.

Example (using SAS):
Triangle ABC and triangle DEF have AB = DE, AC = DF, and angle A equals angle D (the included angle). Then triangles are congruent by SAS, so BC = EF as well.

Common pitfall: claiming SAS when the given angle is not included between the two sides.

Similar figures (dilations and scale factors)

Two figures are similar if corresponding angles are equal and corresponding side lengths are proportional.

The constant multiplier between corresponding sides is the scale factor k.

If two triangles are similar and one side ratio is known, all side ratios match:

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

Areas scale by the square of the scale factor:

\frac{\text{Area}1}{\text{Area}2} = k^2

Volumes (for similar solids) scale by the cube:

\frac{\text{Vol}1}{\text{Vol}2} = k^3

Why it matters on the ACT: Similarity is a fast way to find unknown lengths without heavy algebra.

Triangle similarity shortcuts

You often don’t need all sides/angles:

• AA: two angles equal implies triangles are similar
• SAS similarity: included angle equal and surrounding side ratio equal
• SSS similarity: all three side ratios equal

Example (AA similarity):
One triangle has angles 40 and 60 degrees. Another has angles 40 and 60 degrees. They are similar (AA). If the scale factor from the first to the second is 3, then every length triples and every area becomes 9 times larger.

Common pitfall: using similarity when triangles are actually congruent (which is fine), but then forgetting to set k = 1 when you compare side ratios.

Exam Focus

• Typical question patterns:
  • Use similar triangles to find a missing side in a diagram (often with parallel lines creating smaller triangles).
  • Identify whether triangles are congruent or similar based on given marks.
  • Use scale factor to relate perimeters/areas.
• Common mistakes:
  • Setting up proportions with mismatched corresponding sides.
  • Forgetting area scales by k^2 (not k).
  • Assuming triangles are similar because “they look similar” rather than using AA/SAS/SSS.

Properties of Triangles and Pythagorean Theorem

Triangles are central in ACT geometry because many shapes can be decomposed into triangles. The quickest path through a messy diagram is often: find a triangle, use its properties, then rebuild the larger figure.

Triangle angle sum and exterior angles

The sum of a triangle’s interior angles is always 180 degrees.

An exterior angle (formed by extending one side) equals the sum of the two remote interior angles. This is useful when you’re given an outside angle in a diagram.

Triangle inequality

In any triangle with side lengths a, b, c:

a + b > c

(and similarly for other pairs). This shows up in “possible side length” questions.

Special triangles

Two special right triangles are worth knowing because they let you avoid trig.

45-45-90 triangle (isosceles right triangle): if legs are x, hypotenuse is:

x\sqrt{2}

30-60-90 triangle: if the shortest side (opposite 30) is x, then hypotenuse is 2x and longer leg is:

x\sqrt{3}

Common pitfall: swapping which side is x\sqrt{3}. In a 30-60-90 triangle, the longest side is always the hypotenuse 2x.

Pythagorean Theorem

For a right triangle with legs a and b, and hypotenuse c:

a^2 + b^2 = c^2

Why it matters: It connects geometry to algebra. Many coordinate geometry distance problems are “hidden Pythagorean theorem” problems.

Example (basic):
A right triangle has legs 6 and 8. Find the hypotenuse.

6^2 + 8^2 = c^2

36 + 64 = c^2

100 = c^2

c = 10

Pythagorean triples

Some integer right triangles appear frequently:

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17

These scale: if 3,4,5 works, then 6,8,10 works, etc.

Area of a triangle

The most common area formula is base times height:

A = \frac{1}{2}bh

You must use a height perpendicular to the base (not just any side length).

Example (area with a perpendicular height):
A triangle has base 10 and height 7. Then:

A = \frac{1}{2}\cdot 10 \cdot 7 = 35

Common pitfall: using a slanted side as the height when it is not perpendicular.

Exam Focus

• Typical question patterns:
  • Find missing sides using Pythagorean theorem (sometimes embedded in rectangles or isosceles triangles).
  • Use special right triangles to compute exact lengths.
  • Use triangle angle sum/exterior angle relationships to solve for x.
• Common mistakes:
  • Squaring incorrectly or forgetting to take the square root at the end.
  • Treating the hypotenuse as a leg (the hypotenuse is always opposite the right angle and is longest).
  • Misidentifying which side corresponds to x\sqrt{3} in a 30-60-90 triangle.

Trigonometric Ratios and Right Triangles

Trigonometry is a way to convert between angles and side ratios in right triangles. On the ACT, trig usually appears as “given an angle and one side, find another side” or “find an angle from side ratios.”

The idea behind sine, cosine, tangent

In a right triangle, pick an acute angle \theta. Relative to that angle:

• Opposite side: across from \theta
• Adjacent side: next to \theta (but not the hypotenuse)
• Hypotenuse: longest side, opposite the right angle

The basic ratios are:

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Why it matters: Similar triangles guarantee that for a fixed angle \theta, these ratios are constant. That’s what makes trig work.

Using trig to solve for sides

You choose the ratio that connects what you know to what you want.

Example (solve for a missing side):
A right triangle has angle \theta = 30 degrees. The hypotenuse is 12. Find the side opposite \theta.

Use sine because it involves opposite and hypotenuse:

\sin(30) = \frac{\text{opposite}}{12}

If you know \sin(30) = \frac{1}{2}, then:

\frac{1}{2} = \frac{\text{opposite}}{12}

\text{opposite} = 6

If you don’t remember exact trig values, the ACT may provide a table or choose values that simplify. Also note that 30 degrees suggests a 30-60-90 triangle, so opposite should be half the hypotenuse.

Inverse trig (finding angles)

If you are given side lengths and need an angle, you set up a ratio and use an inverse function:

\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)

On calculator-allowed sections, this is practical, but many ACT problems avoid heavy inverse trig.

Angle of elevation/depression

These are modeling situations where you interpret a right triangle from a real context.

• Angle of elevation: you look up from horizontal.
• Angle of depression: you look down from horizontal.

The key is that the angle is measured from a horizontal line, so you often get a right triangle with a clear “opposite” (vertical change) and “adjacent” (horizontal distance).

Common pitfall: using the wrong reference angle (mixing up which acute angle is \theta).

Exam Focus

• Typical question patterns:
  • Use \sin, \cos, or \tan to find a missing side given an angle.
  • Use right-triangle trig in word problems (ladders, shadows, building heights).
  • Combine trig with Pythagorean theorem (find one leg, then the other).
• Common mistakes:
  • Confusing opposite and adjacent (label relative to the chosen angle).
  • Using degrees vs radians incorrectly (ACT trig is in degrees).
  • Treating the hypotenuse as adjacent.

Properties of Circles

Circle geometry mixes algebra, angles, and sometimes coordinate geometry. A circle is the set of all points a fixed distance (the radius) from a center.

Radius, diameter, circumference, and area

If radius is r, diameter is:

d = 2r

Circumference:

C = 2\pi r

Area:

A = \pi r^2

Why it matters: Many multi-step problems hide the radius in another measurement (like diameter or a chord).

Central angles, arcs, and sectors

A central angle has its vertex at the center. The measure of an arc (in degrees) matches the measure of its corresponding central angle.

A sector is a “slice” of a circle. If a sector has central angle \theta degrees, then its area is the fraction \theta/360 of the full circle:

A_{sector} = \frac{\theta}{360}\pi r^2

Similarly, arc length is the same fraction of circumference:

L_{arc} = \frac{\theta}{360}2\pi r

Example (sector area):
A circle has r = 6 and a sector angle 90 degrees. Then:

A_{sector} = \frac{90}{360}\pi\cdot 6^2

A_{sector} = \frac{1}{4}\pi\cdot 36 = 9\pi

Common pitfall: using \theta/180 instead of \theta/360 when working in degrees.

Chords and perpendicular bisectors

A chord is a segment with endpoints on the circle.

Key fact: A radius (or diameter) perpendicular to a chord bisects the chord. This is often used to form right triangles inside the circle.

Inscribed angles

An inscribed angle has its vertex on the circle and intercepts an arc.

Core relationship:

• An inscribed angle equals half its intercepted arc.

So if an arc measures 100 degrees, the inscribed angle intercepting it is 50 degrees.

A very high-yield special case: an angle inscribed in a semicircle (intercepting a diameter) is a right angle. In other words, if a triangle has a diameter as one side and the third vertex on the circle, that triangle is right.

Common pitfall: doubling when you should halve (or vice versa). If the vertex is on the circle, think “half.” If the vertex is at the center, think “same as arc.”

Tangents

A tangent touches the circle at exactly one point.

Important fact: A tangent is perpendicular to the radius drawn to the point of tangency. This creates right triangles and triggers Pythagorean theorem.

Another common result: Tangent segments drawn from the same external point to a circle are equal in length.

Exam Focus

• Typical question patterns:
  • Compute arc length/sector area using \theta/360.
  • Use inscribed-angle and diameter-right-angle facts to find angle measures.
  • Use tangent-radius perpendicularity to set up right triangles.
• Common mistakes:
  • Mixing up arc measures and chord lengths (they’re different types of quantities).
  • Forgetting the factor of 1/2 for inscribed angles.
  • Using diameter when the problem gives radius (or vice versa).

Surface Area and Volume of Solids

Solid geometry extends area ideas into 3D. The ACT typically focuses on recognizing formulas and applying them correctly, especially with units.

Prism and cylinder (same structure)

A prism has two parallel congruent bases that are polygons; a cylinder is like a prism with circular bases.

For both, volume is “base area times height”:

V = Bh

Here B is the area of the base, and h is the perpendicular height (distance between bases).

Surface area is “two bases plus lateral area.” For a right prism, lateral area equals perimeter of base times height:

SA = 2B + Ph

For a cylinder with radius r and height h:

V = \pi r^2 h

SA = 2\pi r^2 + 2\pi r h

Common pitfall: confusing lateral area 2\pi r h with total surface area (which includes the two circles).

Pyramid and cone

A pyramid has a polygon base and triangular faces meeting at an apex; a cone is the circular-base version.

Volume is one-third of the corresponding prism/cylinder:

V = \frac{1}{3}Bh

For a cone:

V = \frac{1}{3}\pi r^2 h

Surface area often involves slant height (not the vertical height). For a right circular cone with radius r and slant height \ell:

SA = \pi r^2 + \pi r \ell

Why this matters: the ACT may give you slant height and expect surface area, or give you height and expect volume. Using the wrong height is a common error.

Sphere

A sphere’s formulas are frequently tested.

Surface area:

SA = 4\pi r^2

Volume:

V = \frac{4}{3}\pi r^3

Example (sphere volume):
If r = 3, then:

V = \frac{4}{3}\pi\cdot 3^3 = \frac{4}{3}\pi\cdot 27 = 36\pi

Unit awareness

Areas have squared units (like cm^2) and volumes have cubed units (like cm^3). If a problem mixes units (inches and feet), convert before computing.

Exam Focus

• Typical question patterns:
  • Plug into volume formulas for prisms/cylinders/cones/spheres.
  • Compare volumes or surface areas when a dimension scales (e.g., radius doubles).
  • Find lateral vs total surface area.
• Common mistakes:
  • Using diameter in place of radius.
  • Using slant height in volume (volume uses perpendicular height).
  • Forgetting to include both bases in surface area of prisms/cylinders.

Coordinate Geometry

Coordinate geometry translates shapes into algebra on the coordinate plane. The big advantage is that you can compute distances, slopes, and intersections systematically, even without a diagram.

Distance and midpoint

For points A(x1, y1) and B(x2, y2), the distance formula comes from the Pythagorean theorem:

d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

The midpoint is the average of coordinates:

M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)

Example (distance):
Find the distance between (-1, 2) and (5, -2).

d = \sqrt{(5 - (-1))^2 + (-2 - 2)^2}

d = \sqrt{6^2 + (-4)^2}

d = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}

Common pitfall: forgetting to square the differences before adding, or dropping a negative sign too early.

Slope and line equations

Slope measures steepness:

m = \frac{y2 - y1}{x2 - x1}

Horizontal lines have slope 0; vertical lines have undefined slope.

Common line forms:

• Slope-intercept:

y = mx + b

• Point-slope:

y - y1 = m(x - x1)

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, meaning:

m1 m2 = -1

(when both slopes exist).

Example (perpendicular slope):
If a line has slope \frac{2}{3}, a perpendicular line has slope:

-\frac{3}{2}

Circles on the coordinate plane

A circle with center (h, k) and radius r has equation:

(x - h)^2 + (y - k)^2 = r^2

If the circle is centered at the origin, this simplifies to:

x^2 + y^2 = r^2

Why it matters: Many ACT questions give you the equation and ask for center/radius, or vice versa.

Common pitfall: reading the center as (h, k) instead of (h, k) with signs reversed. In x - h, if you see x + 3, that means h = -3.

Using coordinates to prove geometry facts

You can show a shape is a rectangle by proving adjacent sides are perpendicular (slopes are negative reciprocals) and opposite sides are parallel (slopes equal). You can show it’s a rhombus by proving all sides have equal distance.

Exam Focus

• Typical question patterns:
  • Compute distance, midpoint, or slope from coordinates.
  • Find an equation of a line given slope and point.
  • Extract center/radius from a circle equation.
• Common mistakes:
  • Switching x1 with x2 inconsistently in slope (keep numerator and denominator aligned).
  • Misreading circle centers due to sign errors.
  • Treating vertical lines as having slope 0 (they are undefined).

Equations of Conic Sections

Conic sections are curves formed by slicing a cone, but on the ACT you mainly treat them as graphs and equations. The goal is usually to recognize a conic from its equation and identify key features (center, radius, vertex, axis, etc.).

Circles (a conic you should recognize instantly)

As noted earlier:

(x - h)^2 + (y - k)^2 = r^2

You should be able to read:

• center: (h, k)
• radius: r

Example (identify center and radius):
Given:

(x + 2)^2 + (y - 5)^2 = 16

Center is (-2, 5) and radius is 4.

Parabolas

A parabola is the set of points equidistant from a fixed point (focus) and a line (directrix). On the ACT, you typically use the standard “vertex form” that shows where it is shifted.

Vertical parabola:

y = a(x - h)^2 + k

Horizontal parabola:

x = a(y - k)^2 + h

Here (h, k) is the vertex. The sign of a determines whether it opens up/down (vertical) or left/right (horizontal). The size of a controls how “wide” it is.

Why it matters: Many questions ask for the vertex directly from the equation, or ask how the graph shifts compared to y = x^2.

Common pitfall: thinking k is always positive shift; actually y = (x - h)^2 + k shifts right by h and up by k, with signs embedded.

Ellipses and hyperbolas (recognition more than memorization)

The ACT may test recognition of these by their squared terms.

• Ellipse: both x and y are squared and added, but typically with different coefficients. A common centered-at-origin form is:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

• Hyperbola: both are squared, but one is subtracted. A common centered-at-origin form is:

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

or

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Practical takeaway: plus means ellipse, minus means hyperbola (in these standard forms).

Because ACT problems vary, you’re usually not asked for foci/asymptotes unless the problem provides structure; more often you identify the type or interpret intercepts.

Example (classify):
Classify:

\frac{x^2}{9} + \frac{y^2}{4} = 1

Both squared terms are added, so it’s an ellipse.

Exam Focus

• Typical question patterns:
  • Identify a conic (circle, parabola, ellipse, hyperbola) from its equation.
  • Read vertex/center and basic graph shifts from standard forms.
  • For circles, compute radius/center and possibly intercepts.
• Common mistakes:
  • Sign errors when reading centers in x - h and y - k.
  • Confusing parabola vertex form with factored quadratic form.
  • Assuming any equation with x^2 and y^2 is a circle (it’s only a circle when coefficients match and no xy term).