Geometry & Trigonometry

Strategy/Review

Lines and Angles

Fundamentals:

• Supplementary angles: sum to 180° (linear pair).
• Complementary angles: sum to 90°.
• Vertical angles: equal (opposite angles formed by intersecting lines).
• Parallel lines cut by a transversal: corresponding angles equal, alternate interior angles equal, same-side interior angles supplementary.

Parallel line shortcut: When you see parallel lines with a transversal, all acute angles are equal, all obtuse angles are equal, and acute + obtuse = 180°.

Triangles

Angle Sum: The three angles of a triangle always sum to 180°.

Triangle Inequality: The sum of any two sides must be greater than the third side. This is useful for eliminating impossible side-length combinations.

Types:

• Equilateral: all sides equal, all angles 60°.
• Isosceles: two sides equal, base angles equal.
• Right: one 90° angle. Pythagorean theorem applies.
• Scalene: no equal sides, no equal angles.

Area of a triangle: A = (1/2) × base × height The height must be perpendicular to the chosen base.

The Pythagorean Theorem

Formula: a² + b² = c² (where c is the hypotenuse) Applies ONLY to right triangles.

Pythagorean Triples (memorize these — they appear constantly):

• 3-4-5 (and multiples: 6-8-10, 9-12-15, 12-16-20, 30-40-50)
• 5-12-13 (and multiple: 10-24-26)
• 8-15-17
• 7-24-25

Recognizing these triples saves you from computing square roots. If you see a right triangle with legs 6 and 8, the hypotenuse is 10 (the 3-4-5 triple × 2).

Special Right Triangles

45-45-90 Triangle (isosceles right triangle):

• Angles: 45° − 45° − 90°
• Side ratio: leg : leg : hypotenuse = 1 : 1 : √2
• If leg = x, then hypotenuse = x√2
• If hypotenuse = h, then leg = h/√2 = (h√2)/2

30-60-90 Triangle:

• Angles: 30° − 60° − 90°
• Side ratio: short leg : long leg : hypotenuse = 1 : √3 : 2
• Short leg (opposite 30°) = half the hypotenuse
• Long leg (opposite 60°) = short leg × √3
• If hypotenuse = h, then short leg = h/2, long leg = (h√3)/2

Recognition cues:

• A right triangle with a 45° angle → 45-45-90.
• A right triangle with a 30° or 60° angle → 30-60-90.
• An equilateral triangle split by an altitude → creates two 30-60-90 triangles.

Similar Triangles

Two triangles are similar if their corresponding angles are equal (AA criterion is sufficient). Similar triangles have proportional sides.

Key pattern: A line drawn parallel to one side of a triangle creates a smaller similar triangle.

Example: In triangle ABC, a line DE is parallel to BC. If AD = 3, DB = 6, and DE = 4, find BC. → AD/AB = DE/BC → 3/9 = 4/BC → BC = 12.

Circles

Formulas (r = radius, d = diameter = 2r):

• Circumference: C = 2πr = πd
• Area: A = πr²
• Arc length: L = (θ/360°) × 2πr (where θ is the central angle in degrees)
• Sector area: A_sector = (θ/360°) × πr²

Inscribed angles: An angle inscribed in a semicircle is a right angle (Thales' theorem). An inscribed angle is half the measure of its intercepted arc.

Perimeter and Area of 2D Shapes

Surface Area and Volume of 3D Solids

Trigonometry: SOH CAH TOA

For a right triangle with angle θ:

• Sin θ = Opposite / Hypotenuse
• Cos θ = Adjacent / Hypotenuse
• Tan θ = Opposite / Adjacent

Special Angle Values (MUST MEMORIZE):

Memory aids:

• sin 30° = 1/2 (smallest meaningful angle, smallest sine)
• sin and cos swap for complementary angles: sin(30°) = cos(60°), sin(45°) = cos(45°)
• tan = sin/cos

Applications: Trig on the DAT is almost always about finding a missing side length given an angle and one side.

Example: A 20-foot ladder leans against a wall making a 60° angle with the ground. How high up the wall does it reach? → The height is opposite the 60° angle. Hypotenuse = 20 ft. → sin(60°) = opposite / 20 → √3/2 = h/20 → h = 10√3 ≈ 17.3 ft.

Common Traps

Trap 1 — Using Pythagorean Theorem on Non-Right Triangles. a² + b² = c² ONLY works for right triangles. If no right angle is given, you can't use it directly (need law of cosines, which isn't on the DAT).

Trap 2 — Confusing 30-60-90 and 45-45-90 Ratios. The 30-60-90 has ratio 1:√3:2. The 45-45-90 has ratio 1:1:√2. Mixing them up (e.g., thinking a 45-45-90 has side ratios involving √3) is a common error.

Trap 3 — Forgetting Units in Area/Volume. Area is in square units (cm², m²), volume in cubic units (cm³, m³). The DAT sometimes asks for conversion or comparison of different units. 1 m² = 10,000 cm² (not 100).

Trap 4 — Using Diameter Instead of Radius. Area = πr², not πd². Circumference = 2πr = πd. If the problem gives diameter, halve it before computing area.

Trap 5 — SOH CAH TOA on the Wrong Angle. Identify which angle is θ. Opposite means the side across from that specific angle. Adjacent means the leg touching that angle (not the hypotenuse).