SAT Geometry and Trigonometry: The Definitive Study Guide

SAT Math Section Statistics and Adaptive Modules

• Contextual Distribution: Geometry and trigonometry questions account for approximately $14.2\, \%$ of the SAT math section based on recent statistics.

• Digital SAT Structure: The math section is adaptive, consisting of Module 1 and Module 2.

  • Module 1: Contains a higher volume of geometry and trigonometry questions compared to Module 2.

  • Module 2: If a student performs well in Module 1, they receive the "hard" version of Module 2. This second module contains fewer geometry/trig questions, but the difficulty level is significantly higher.

• Learning Progression: The topic follows a building-block hierarchy: understanding angles and lines is required for triangles, which is necessary for similar shapes, which eventually leads to area and volume calculations.

Angles, Lines, and Triangles

To master this section, students must recognize seven specific patterns. These allow for the calculation of unknown angles without guessing.

• Pattern 1: Vertical Angles: Whenever straight lines intersect, the angles directly across from each other at the intersection are equal.

  • Multiple lines can intersect; as long as the angles are horizontal/vertical opposites at the vertex of two straight lines, they are identical.

• Pattern 2: Supplementary Angles: A straight line has an angle of $180^{\circ}$. Therefore, any group of angles resting on a straight line must sum to $180^{\circ}$.

• Pattern 3: Corresponding Angles: This pattern applies only to paral0lel lines (indicated by arrow marks or explicit text in the question).

  • If a line intersects two parallel lines, the angles in the same relative position of each intersection are equal (e.g., top-right to top-right).

• Pattern 4: Alternate Interior Angles: For parallel lines, angles on opposite sides of the transversal and inside the parallel lines are equal.

  • This can be derived by combining corresponding and vertical angle rules (the "Z" pattern).

• Pattern 5: Triangle Angle Sum: All interior angles in any triangle must sum to $180^{\circ}$. If two angles are known, the third is found by $180 - (\text{angle}{1} + \text{angle}{2})$.

• Pattern 6: Isosceles Triangles: Occurs when two sides or two angles are equal (often marked with single or double slashes/arcs).

  • Knowing just one angle in an isosceles triangle allows for the derivation of the other two using the sum to $180^{\circ}$ rule and dividing remaining values by two where applicable.

• Pattern 7: Equilateral Triangles: All sides and all angles are equal. Since they must sum to $180^{\circ}$, each angle is always exactly $60^{\circ}$ ($180 / 3 = 60$).

Polygon Calculations and Strategies

• Sum of Interior Angles: The formula for the interior angle sum of a polygon with $n$ sides is $180 \times (n - 2)$.

  • Example: A polygon with $99$ sides sums to $180 \times (99 - 2) = 17,460^{\circ}$.

• Strategy for Complex Figures: Always look for triangles within complex line intersections. Work backward from the target angle, using vertical, supplementary, and triangle sum rules to fill in the "puzzle" pieces.

Right Triangles and Trigonometry

Specific rules for right triangles (triangles with a $90^{\circ}$ angle) include SOHCAHTOA, the Pythagorean Theorem, and Special Right Triangles.

• SOHCAHTOA (Trigonometric Ratios): An acronym for defining sine, cosine, and tangent in relation to a specific angle ($x$).

  • $\sin(x)$: Opposite / Hypotenuse.

  • $\cos(x)$: Adjacent / Hypotenuse.

  • $\tan(x)$: Opposite / Adjacent.

  • Definitions: The "Opposite" side is the one furthest from the angle. "Adjacent" is the side next to the angle. "Hypotenuse" is the longest side, always across from the $90^{\circ}$ angle.

• Trigonometric Proportions: Trig values represent ratios, not necessarily absolute lengths. A $\sin(x) = \frac{4}{5}$ could mean sides are $4$ and $5$, or $8$ and $10$, or $4x$ and $5x$.

• Pythagorean Theorem: Used to find a missing side length when two other sides are known: $a^{2} + b^{2} = c^{2}$.

  • Common triplets include $3$-$4$-$5$ and its multiples ($6$-$8$-$10$).

• Special Right Triangles: These allow solving for sides with only one known length.

  • $30^{\circ} \text{--} 60^{\circ} \text{--} 90^{\circ}$: Sides follow the ratio $x : x\sqrt{3} : 2x$. A dead giveaway is seeing $\sqrt{3}$ in the question or answers. These are often formed by bisecting an equilateral triangle.

  • $45^{\circ} \text{--} 45^{\circ} \text{--} 90^{\circ}$: Sides follow the ratio $x : x : x\sqrt{2}$. Dead giveaways include seeing $\sqrt{2}$ or an isosceles right triangle. These are formed by cutting a square diagonally.

• Complementary Rule: $\sin(x) = \cos(90 - x)$ and $\cos(x) = \sin(90 - x)$. This serves as a shortcut for right triangles where the two non-right angles must add up to $90^{\circ}$.

Similar Shapes and Proportions

Similar shapes have the same proportions and angles, but different absolute sizes. One is a "stretched" version of the other.

Proving Similarity

• Triangles: Similarity is proven if two angles match (AA) or if all three sides are proportional (SSS).

• Multiple Triangles Rule: If a question includes multiple triangles, they are almost certainly similar.

Four-Step Process for Similar Triangle Questions

1. Draw Identically: Redraw the triangles separately so they are oriented the same way.

2. Label Angles: Use vertical angles or given text to identify which angle corresponds to which. Avoid assuming orientation based on appearance.

3. Label Known Sides: Transfer all numerical data to the new drawings.

4. Set Up Proportions: Use a single variable equation (e.g., $\frac{\text{Side}_{\text{Big}}}{\text{Side}_{\text{Small}}} = \frac{\text{OtherSide}_{\text{Big}}}{\text{OtherSide}_{\text{Small}}}$) and solve.

The Proportion Trick (Scale Factors)

• If the side length scale factor between two similar shapes is $k$:

  • The Area (or Surface Area) scale factor is $k^{2}$.

  • The Volume scale factor is $k^{3}$.

• Example: If the side length of a cube is doubled ($k=2$), its surface area increases by $4$ ($2^{2}$) and its volume increases by $8$ ($2^{3}$).

• Hard Application: If given that an area is $25$ times larger, first find the side scale factor ($\sqrt{25} = 5$), then use that to find the volume increase ($5^{3} = 125$).

Area and Volume Formulas and Conceptual Applications

Students have access to the SAT reference sheet, but conceptual understanding is necessary for harder variations.

• Rectangle Area: $L \times W$. Conceptually, this is stacking a line of length $L$, $W$ times.

• Rectangular Prism Volume: $L \times W \times H$ (or Base Area $\times H$). Conceptually, this is stacking rectangles on top of each other.

• Cylinder Volume: $\pi \times r^{2} \times h$. This is a stack of circular bases ($\pi r^{2}$).

• Surface Area Conceptualization:

  • Rectangular Prism: Sum of all six rectangular faces.

  • Cylinder: The area of two circles ($2\pi r^{2}$) plus the area of the "unrolled" side rectangle ($2\pi r \times h$).

  • Pyramid (Rectangular or Triangular): The base area plus the area of the triangular faces.

  • Pitfall: Do not use the height of the pyramid ($H$) to calculate the area of the triangular side faces. You must use the "slant height," found by creating a right triangle between the pyramid's center, the center of an edge, and the tip.

• Shaded Region Strategy: Calculate the area of the entire shape and subtract the area of the unneeded internal shape.

• Inscribed Shapes: When one shape fits perfectly inside another (e.g., a rectangle inside a circle).

  • Strategy: Find the common length. For a rectangle in a circle, the diagonal of the rectangle is the same as the diameter of the circle.

Proving Congruency and Similarity

Typically found in the Hard Module 2, these involve proving triangles are identical or proportional.

• Similarity Proofs: AA (Angle-Angle) or SSS (proportional sides).

• Congruency Proofs (The "Identical" versions):

  • SSS: All sides are equal.

  • SAS: Two sides and the angle between them are equal.

  • ASA: Two angles and the side between them are equal.

  • AAS: Two angles and a side not between them are equal.

  • Forbidden Rules: No "AAA" (proves similarity only) and No "SSA" (the "Anti-PG" rule).

• Triangle Inequality Theorem: The sum of any two sides of a triangle must always be strictly greater than the third side (a + b > c).

  • If sides are $7$ and $10$, the third side $x$ must be: 10 - 7 < x < 10 + 7 \rightarrow 3 < x < 17.

Circle Characteristics and Arc/Sector Formulas

• The Pizza Slice Formula: The central angle ($n$), arc length ($L$), and sector area ($A$) are proportional to the whole circle:

  • $\frac{n}{360} = \frac{L}{2\pi r} = \frac{A}{\pi r^{2}}$

• Tangent Line Property: A line tangent to a circle is always perpendicular ($90^{\circ}$) to the radius at the point of contact.

• Radii and Isosceles Triangles: Any triangle formed by drawing two radii from the center to the circle's edge is an isosceles triangle because the radii are equal lengths.

• Inscribed Angles: If a central angle and an angle on the circle's circumference share the same endpoints, the angle on the circumference is always exactly half the size of the central angle.