Geometry and Trigonometry

Geometry: Lines, Angles, and Triangles

Part 1 of 7 — Angle Relationships

Geometry accounts for roughly 10-15% of SAT Math questions. Mastering angle relationships gives you quick points.

Fundamental Angle Rules

• Supplementary angles: Sum to $180°$
• Complementary angles: Sum to $90°$
• Vertical angles: Equal (formed by intersecting lines)
• Angles on a straight line: Sum to $180°$

Parallel Lines Cut by a Transversal

When a line crosses two parallel lines, it creates 8 angles with key relationships:

• Corresponding angles are equal (same position at each intersection)
• Alternate interior angles are equal (opposite sides, between parallels)
• Alternate exterior angles are equal (opposite sides, outside parallels)
• Co-interior (same-side interior) angles sum to $180°$

Triangle Angle Sum

The angles in any triangle sum to $180°$.

If a triangle has angles $55°$ and $70°$: $\text{Third angle} = 180° - 55° - 70° = 55°$

This is an isosceles triangle (two equal angles → two equal sides).

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

$\text{Exterior angle} = \text{Remote interior}_1 + \text{Remote interior}_2$

SAT Trap ⚠️

When the SAT shows a figure with parallel lines, check if they actually SAY the lines are parallel. "Looks parallel" ≠ IS parallel. Look for arrows or explicit statements.

Angle Relationships Practice 🎯

Deep Dive: Multi-Step Angle Problems

Worked Example 1: Parallel Lines with Algebra

Worked Example 2: Exterior Angle with Algebra

Angle Relationship Quick Reference

Advanced Angle Problems 🎯

Identify the Angle Relationship — Name each angle pair.

Part 1 Summary: Angle Relationships

SAT Strategy

• If angles involve variables, set up an equation using the appropriate rule.
• Always check: do the angles sum correctly?
• Angles at a point sum to $360°$ — don't confuse with straight line ($180°$).

Next: Triangle properties, special right triangles, and similarity →

Triangle Properties & Theorems

Part 2 of 7 — Special Triangles, Similarity, Congruence

Special Right Triangles

The SAT provides these in the reference sheet, but memorizing them saves time:

45-45-90 Triangle:

• Legs: $x$, $x$
• Hypotenuse: $x\sqrt{2}$

30-60-90 Triangle:

• Short leg: $x$ (opposite 30°)
• Long leg: $x\sqrt{3}$ (opposite 60°)
• Hypotenuse: $2x$ (opposite 90°)

Example

A 30-60-90 triangle has a hypotenuse of 10. Find the legs.

• Hypotenuse $= 2x = 10$ → $x = 5$
• Short leg $= 5$
• Long leg $= 5\sqrt{3} \approx 8.66$

Triangle Inequality Theorem

For any triangle with sides $a$, $b$, $c$: $a + b > c$

The sum of any two sides must exceed the third.

Example: Can a triangle have sides 3, 5, and 9?
$3 + 5 = 8 < 9$ → No!

Similar Triangles (AA Similarity)

If two angles of one triangle equal two angles of another, the triangles are similar (same shape, proportional sides).

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

SAT Trap ⚠️

In 30-60-90 triangles, students often mix up which leg is which. Remember: the shortest side is opposite the smallest angle (30°).

Triangle Properties Practice 🎯

Deep Dive: Applying Triangle Properties

Worked Example 1: Special Triangle → Area

SAT shortcut: Area of equilateral triangle $= \frac{s^2\sqrt{3}}{4}$. Plug in: $\frac{64\sqrt{3}}{4} = 16\sqrt{3}$ ✓

Worked Example 2: Similar Triangles with Algebra

Pythagorean Triples to Memorize

Recognizing these saves you from using the Pythagorean theorem every time.

Triangle Inequality: Finding the Range

If two sides are $5$ and $11$, the third side $x$ must satisfy: $11 - 5 < x < 11 + 5$ $6 < x < 16$

Advanced Triangle Problems 🎯

Match the Triangle Property — Select the correct value for each scenario.

Part 2 Summary: Triangle Properties

SAT Strategy

• Spot Pythagorean triples (3-4-5, 5-12-13) before computing.
• For special right triangles, identify which angle or side you're given first, then find $x$.
• In similar triangle problems, match corresponding sides carefully — order matters.

Next: Area, perimeter, and quadrilateral properties →

Area, Perimeter, and Quadrilaterals

Part 3 of 7 — Polygons and Their Properties

Essential Area Formulas

Key Insight: Height ≠ Side Length

The height (altitude) is the perpendicular distance from base to top. In non-right triangles and parallelograms, the height is NOT the same as a side length.

Coordinate Geometry Areas

For a rectangle or right triangle on the coordinate plane:

• Find the lengths of the sides using the distance formula or by counting grid units
• Apply the appropriate area formula

Shaded Region Problems

Strategy: $\text{Shaded area} = \text{Total area} - \text{Unshaded area}$

Example: A circle of radius 5 is inscribed in a square. Find the shaded area (corners).

• Square area: $(2 \times 5)^2 = 100$
• Circle area: $\pi(5)^2 = 25\pi \approx 78.54$
• Shaded area: $100 - 25\pi \approx 21.46$

SAT Trap ⚠️

In "shaded region" problems, make sure you subtract the RIGHT shape. Draw the overlapping shapes clearly and label dimensions.

Area & Perimeter Practice 🎯

Deep Dive: Complex Area Problems

Worked Example 1: Multi-Shape Shaded Region

Worked Example 2: Coordinate Plane Area

Coordinate area shortcut: When one side is horizontal or vertical, use it as the base — the height is just the perpendicular distance.

Height vs. Slant Side — The #1 Trap

SAT trap: A parallelogram has sides 8 and 5 with a height of 4. Area $= 8 \times 4 = 32$ (NOT $8 \times 5 = 40$).

Advanced Area & Perimeter Problems 🎯

Choose the Right Formula — Select the correct area formula for each shape.

Part 3 Summary: Area & Perimeter

Key Strategies

• Shaded regions: Total area − unshaded area. Draw and label clearly.
• Coordinate plane: Use horizontal/vertical sides as base when possible.
• Height trap: Always use the PERPENDICULAR height, not the slant side.

Next: Circle geometry — arcs, sectors, and central angles →

Circles: Arc Length, Sector Area, Central Angles

Part 4 of 7 — Circle Geometry

Circle Fundamentals

Where $\theta$ is the central angle in degrees.

The Proportion Rule

A central angle of $\theta°$ creates an arc that is $\frac{\theta}{360}$ of the full circle. This fraction applies to BOTH arc length AND sector area.

Example: A circle with radius 10 has a central angle of $72°$.

• Arc length $= \frac{72}{360} \times 2\pi(10) = \frac{1}{5} \times 20\pi = 4\pi$
• Sector area $= \frac{72}{360} \times \pi(10)^2 = \frac{1}{5} \times 100\pi = 20\pi$

Inscribed Angle Theorem

An inscribed angle is HALF the central angle that subtends the same arc.

$\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}$

Special case: An inscribed angle that subtends a diameter (semicircle) is always $90°$.

Tangent Lines

A tangent to a circle is perpendicular to the radius at the point of tangency, so the radius and the tangent line form a right angle there.

Circle Geometry Practice 🎯

Deep Dive: Multi-Step Circle Problems

Worked Example 1: Arc Length from Context

Worked Example 2: Sector Area to Find Radius

Key Circle Relationships

Radians on the SAT

Some SAT questions use radians instead of degrees:

• Full circle $= 2\pi$ radians $= 360°$
• Arc length in radians: $L = r\theta$
• Sector area in radians: $A = \frac{1}{2}r^2\theta$

Conversion: $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$

Advanced Circle Problems 🎯

Circle Calculations — Select the correct result.

Part 4 Summary: Circle Geometry

SAT Strategy

• The fraction $\frac{\theta}{360}$ is the same for both arc length and sector area.
• If the SAT gives you arc length, work backward to find radius or angle.
• Watch for radian vs. degree — the formulas change.

Next: Volume and surface area of 3D figures →

Volume and Surface Area

Part 5 of 7 — 3D Figures

The SAT reference sheet includes these formulas, but knowing them cold saves time.

Volume Formulas

Surface Area

Common SAT Problem: Filling and Draining

"A cylindrical tank has radius 3 ft and height 10 ft. Water fills it at 2 cubic feet per minute. How long until it's full?"

$V = \pi(3)^2(10) = 90\pi \approx 282.7 \text{ ft}^3$ $\text{Time} = \frac{90\pi}{2} = 45\pi \approx 141.4 \text{ minutes}$

Scaling Rule for 3D

If dimensions are scaled by factor $k$:

• Lengths scale by $k$
• Areas scale by $k^2$
• Volumes scale by $k^3$

Example: If you double all dimensions of a box, its volume increases by $2^3 = 8$ times.

Volume & Surface Area Practice 🎯

Deep Dive: 3D Problem-Solving Strategies

Worked Example 1: Transferring Between Shapes

Worked Example 2: Surface Area in Context

Scaling Rules — Complete Table

Common SAT 3D Question Types

1. "How much fits inside?" → Volume
2. "How much material to cover?" → Surface area
3. "Pour from one to another" → Set volumes equal
4. "What happens when dimensions change?" → Scaling rules
5. "How long to fill/drain?" → Volume \div rate

Advanced Volume & Surface Area 🎯

3D Figure Identification — Match the description to the correct formula or value.

Part 5 Summary: Volume & Surface Area

Scaling Rules

• Lengths $\times k$, Areas $\times k^2$, Volumes $\times k^3$

SAT Strategy

• "Pour from shape A to shape B" → Set volumes equal and solve.
• Watch units — if radius is in cm and height in m, convert first.
• Cone = $\frac{1}{3}$ cylinder; hemisphere = $\frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$.

Next: Coordinate geometry — distance, midpoint, and circle equations →

Coordinate Geometry

Part 6 of 7 — Distance, Midpoint, and Equations of Lines/Circles

Distance Formula

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This is just the Pythagorean theorem applied to the coordinate plane.

Midpoint Formula

$M = \left(\frac{x_1 + x_2}{2},\, \frac{y_1 + y_2}{2}\right)$

Slope

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

Parallel lines: Same slope ($m_1 = m_2$)
Perpendicular lines: Negative reciprocal slopes ($m_1 \times m_2 = -1$)

Equation of a Circle

Standard form: $(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$
• Radius: $r$

Example: $(x - 3)^2 + (y + 2)^2 = 25$

• Center: $(3, -2)$ ← note: $y + 2$ means $k = -2$
• Radius: $\sqrt{25} = 5$

Converting General Form to Standard Form (Completing the Square)

$x^2 + y^2 - 6x + 4y - 12 = 0$

Group and complete the square: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$ $(x - 3)^2 + (y + 2)^2 = 25$

SAT Trap ⚠️

When reading circle equations, remember: $(x - h)^2$ means the center's x-coordinate is $+h$, and $(y + k)^2$ means the center's y-coordinate is $-k$. The signs flip!

Coordinate Geometry Practice 🎯

Deep Dive: Coordinate Geometry Problem Solving

Worked Example 1: Finding a Missing Vertex

Worked Example 2: Is It a Right Triangle?

Equation of a Line — Forms You Need

Perpendicular Bisector Strategy

To find the perpendicular bisector of segment $\overline{AB}$:

1. Find the midpoint of $AB$
2. Find the slope of $AB$
3. Take the negative reciprocal for the perpendicular slope
4. Write the line through the midpoint with that slope

Advanced Coordinate Geometry 🎯

Coordinate Geometry Quick Checks — Select the correct answer.

Part 6 Summary: Coordinate Geometry

SAT Strategy

• Know your Pythagorean triples — saves time on distance problems.
• Completing the square converts general form circles to standard form.
• Watch the sign flip in circle equations: $(x + 3)^2$ means center $x = -3$.

Next: Comprehensive geometry review and SAT strategy →

Geometry Review & SAT Strategy

Part 7 of 7 — Comprehensive Review

Formula Quick Reference

Common SAT Geometry Question Patterns

1. "Find the missing angle" → Use angle sum rules
2. "Find the area of the shaded region" → Total minus unshaded
3. "Similar triangles" → Set up proportions
4. "Volume word problem" → Identify the shape, plug into formula
5. "Coordinate geometry" → Distance, midpoint, or circle equation

Strategy: Draw It

If the SAT doesn't give you a figure, draw one yourself. Even a rough sketch helps you avoid errors.

If they DO give you a figure:

• "Not drawn to scale" → Don't trust visual proportions
• "Figure drawn to scale" → You can estimate to eliminate wrong answers

Top 3 Geometry Mistakes

1. Using the wrong formula (mixing up circumference and area)
2. Forgetting to take the square root when finding radius from area
3. Not converting units (e.g., diameter given but formula needs radius)

Geometry Comprehensive Review 🎯

Deep Dive: Multi-Step SAT Geometry Problems

Worked Example 1: Combining Multiple Concepts

Worked Example 2: Coordinate + Geometry Hybrid

SAT Geometry Decision Framework

Common SAT Geometry Mistakes — Quick Check

SAT Geometry Challenge 🎯

Geometry Concept Quick Check — Select the correct answer for each scenario.

Full Topic Summary: Geometry & Angles

Top SAT Geometry Strategies

1. Draw and label — if no figure given, sketch one
2. Know your triples — 3-4-5, 5-12-13, 8-15-17
3. Height ≠ slant side — always perpendicular
4. Diameter vs. radius — read carefully, divide by 2 if needed
5. "Not drawn to scale" — don't trust the picture
6. Complete the square for circle equations in general form
7. Scaling: lengths $k$, areas $k^2$, volumes $k^3$

🎉 Geometry & Angles complete! You're ready for SAT geometry questions.