Plane Geometry (Angles, Triangles, Circles)

Angle Fundamentals

Types of angles:

• Acute: less than $90°$
• Right: exactly $90°$ (marked with a small square)
• Obtuse: between $90°$ and $180°$
• Straight: exactly $180°$ (a straight line)
• Reflex: between $180°$ and $360°$

Angle pair relationships:

• Complementary angles sum to $90°$. If one angle is $35°$, its complement is $55°$.
• Supplementary angles sum to $180°$. If one angle is $110°$, its supplement is $70°$.
• Vertical angles are the non-adjacent angles formed when two lines intersect. Vertical angles are always equal. If two lines cross and one angle is $65°$, the angle directly across is also $65°$, and the other two angles are each $115°$.

Angles with Parallel Lines and Transversals

When a transversal (a line that crosses two parallel lines), it creates eight angles with specific, predictable relationships. This is one of the most frequently tested concepts on the ACT.

• Corresponding angles are equal (same position at each intersection — both upper-left, both lower-right, etc.)
• Alternate interior angles are equal (opposite sides of the transversal, between the parallel lines — these form a "Z" shape)
• Alternate exterior angles are equal (opposite sides, outside the parallel lines)
• Co-interior angles (same-side interior) are supplementary (sum to $180°$ — same side of transversal, between the lines)

Key insight: If you know any ONE of the eight angles, you can determine ALL eight. Each angle is either equal to the known angle or supplementary to it.

Triangle Properties

Angle sum theorem: The three interior angles of any triangle always sum to exactly $180°$. This is perhaps the single most-used geometry fact on the ACT.

Exterior angle theorem: An exterior angle of a triangle (formed by extending one side) equals the sum of the two non-adjacent interior angles. If the interior angles are $40°$, $60°$, and $80°$, the exterior angle adjacent to the $80°$ angle is $40° + 60° = 100°$.

Types of triangles by sides:

• Equilateral: All three sides equal → all three angles are $60°$
• Isosceles: Two sides equal → the base angles (opposite the equal sides) are equal
• Scalene: All sides different → all angles different

Types of triangles by angles:

• Acute: All angles less than $90°$
• Right: One angle equals $90°$
• Obtuse: One angle greater than $90°$

Triangle inequality theorem: The sum of any two sides must be greater than the third side. For sides $a$, $b$, $c$: $a + b > c$, $a + c > b$, and $b + c > a$. The ACT uses this to ask which set of side lengths can form a triangle.

The Pythagorean Theorem

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

$a^2 + b^2 = c^2$

Common Pythagorean triples (memorize these — they save enormous time):

• $3, 4, 5$ (and multiples: $6, 8, 10$; $9, 12, 15$; $12, 16, 20$; $15, 20, 25$)
• $5, 12, 13$ (and multiples: $10, 24, 26$)
• $8, 15, 17$
• $7, 24, 25$

Recognizing a Pythagorean triple instantly eliminates the need for square root calculations. When you see a right triangle with legs 9 and 12, you should immediately know the hypotenuse is 15 (the $3$-$4$-$5$ triple scaled by 3).

Converse: If $a^2 + b^2 = c^2$ for three sides, the triangle is a right triangle. If $a^2 + b^2 > c^2$, it is acute. If $a^2 + b^2 < c^2$, it is obtuse.

Special Right Triangles

These appear on virtually every ACT and must be memorized:

45-45-90 triangle: Sides in ratio $1 : 1 : \sqrt{2}$.

Both legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg. This triangle results from cutting a square along its diagonal. If each leg is $s$, the hypotenuse is $s\sqrt{2}$. If the hypotenuse is $h$, each leg is $\frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2}$.

30-60-90 triangle: Sides in ratio $1 : \sqrt{3} : 2$.

The side opposite $30°$ is the shortest. The side opposite $60°$ is $\sqrt{3}$ times the shortest side. The hypotenuse (opposite $90°$) is exactly twice the shortest side. This triangle results from cutting an equilateral triangle in half.

Similar Triangles

Two triangles are similar if their corresponding angles are equal. This can be established by:

• AA (Angle-Angle): If two angles of one triangle equal two angles of another, the triangles are similar. (The third angles must also be equal, since angles sum to $180°$.)

When triangles are similar:

• Corresponding sides are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$ (the scale factor)
• The ratio of their perimeters equals the scale factor $k$
• The ratio of their areas equals the square of the scale factor: $\frac{A_1}{A_2} = k^2$

This last point is crucial: if the sides are in ratio $2:3$, the areas are in ratio $4:9$.

Congruence

Two triangles are congruent if they have the same shape AND size. Methods to prove congruence: SSS, SAS, ASA, AAS, HL (for right triangles). The ACT tests congruence less frequently than similarity.

Area Formulas

Shape	Area Formula	Notes
Triangle	$A = \frac{1}{2}bh$	$h$ is the perpendicular height
Rectangle	$A = lw$
Square	$A = s^2$	Also $A = \frac{d^2}{2}$ where $d$ is diagonal
Parallelogram	$A = bh$	$b$ is base, $h$ is height
Trapezoid	A = rac{1}{2}(b_1 + b_2)h	Average of parallel sides $imes$ height
Circle	$A = pi r^2$	Radius $r$

Perimeter and Circumference

• Perimeter of a polygon: Sum of all side lengths.
• Circumference of a circle: $C = 2pi r$ or $C = pi d$.

Note: The ACT often asks for the perimeter of complex shapes composed of rectangles and semicircles. Break them down part by part.

Circle Theorems

Central angle: An angle formed by two radii with the vertex at the center. The measure of the central angle equals the measure of its intercepted arc.

Inscribed angle: An angle formed by two chords with the vertex on the circle. The measure of the inscribed angle is half the measure of its intercepted arc. ext{Inscribed Angle} = rac{1}{2} imes ext{Intercepted Arc}

Thales's Theorem: Any triangle inscribed in a circle where one side is a diameter is a right triangle. The angle opposite the diameter is $90°$.

Tangent line: A line tangent to a circle is perpendicular to the radius drawn to the point of tangency. This creates right triangles, which allows you to use the Pythagorean theorem.

Arc length: A portion of the circumference. ext{Arc Length} = rac{ heta}{360°} imes 2pi r where $heta$ is the central angle in degrees.

Sector area: A "slice of pie" portion of the circle's area. ext{Sector Area} = rac{ heta}{360°} imes pi r^2

Polygons

Sum of interior angles: For a polygon with $n$ sides, the sum of the interior angles is: $ext{Sum} = (n-2) imes 180°$

Regular polygon: A polygon where all sides and all angles are equal. Each interior angle measures: ext{Angle} = rac{(n-2) imes 180°}{n}

Exterior angles: The sum of the exterior angles of any convex polygon is always $360°$.

Tip 1: Draw and Label Everything

Geometry is visual. If a problem describes a shape without showing it, draw it immediately. If a figure is provided, label every known length and angle directly on the drawing. Mark right angles with squares and equal sides with tick marks. This reveals relationships that are impossible to see in your head.

Tip 2: Look for Hidden Triangles

Complex shapes are often just triangles in disguise. A trapezoid can be split into a rectangle and two right triangles. A regular hexagon is composed of six equilateral triangles. When in doubt, draw lines to create triangles — especially right triangles, because then you can use the Pythagorean theorem.

Tip 3: Trust the Figure (Mostly)

On the ACT (unlike the SAT), figures are drawn to scale unless explicitly stated "Not drawn to scale." You can use this to your advantage. If an angle looks obtuse, it probably is. If one line looks twice as long as another, it probably is. Use this to estimate and eliminate obviously wrong answer choices.

Tip 4: Memorize the Formulas

Since there is no formula sheet, you must memorize the area formulas (especially trapezoid and circle) and the Pythagorean triples. Writing them down at the start of the section can help relieve anxiety.

Tip 5: Work Backwards from the Answer

In geometry problems asking for a specific length or angle, you can often plug in the answer choices. If the question asks for the width of a rectangle, try the values in the choices to see which one produces the correct area or perimeter given in the problem.