🎯⭐ INTERACTIVE LESSON

Geometry and Trigonometry

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Geometry and Trigonometry - Complete Interactive Lesson

Part 1: Lines & Angles

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 🎯

Part 2: Triangle Properties

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: $x\sqrt{2}$

Part 3: Circle Properties

Area, Perimeter, and Quadrilaterals

Part 3 of 7 — Polygons and Their Properties

Essential Area Formulas

Shape	Area	Perimeter
Rectangle	$A = lw$	$P = 2l + 2w$

Part 4: Area & Volume

Circles: Arc Length, Sector Area, Central Angles

Part 4 of 7 — Circle Geometry

Circle Fundamentals

Property	Formula
Circumference	$C = 2\pi r = \pi d$
Area	$A = \pi r^2$

Part 5: Coordinate Geometry

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

Shape	Volume
Rectangular prism	$V = lwh$
Cylinder	$V = \pi r^2 h$

Part 6: Problem-Solving Workshop

Coordinate Geometry

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

Distance Formula

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

Part 7: Review & Applications

Geometry Review & SAT Strategy

Part 7 of 7 — Comprehensive Review

Formula Quick Reference

Category	Key Formulas
Angles	Triangle sum $= 180°$ , exterior angle $=$ sum of remotes
Triangles	$A = \frac{1}{2}bh$ , Pythagorean theorem:

30-60-90 Triangle:

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

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).

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

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 🎯

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: $100 - 25\pi \approx 21.46$

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

Area & Perimeter Practice 🎯

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

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 $90°$ .

A tangent to a circle is perpendicular to the radius at the point of tangency (

Circle Geometry Practice 🎯

Shape	Surface Area
Rectangular prism	$SA = 2(lw + lh + wh)$
Cylinder	$SA = 2\pi r^2 + 2\pi rh$
Sphere	$SA = 4\pi r^2$

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?"

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

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 🎯

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

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

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

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: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$ $(x - 3)^2 + (y + 2)^2 = 25$

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 🎯

Common SAT Geometry Question Patterns

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

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

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

Geometry Comprehensive Review 🎯

Geometry and Trigonometry

Example

Triangle Inequality Theorem

Similar Triangles (AA Similarity)

SAT Trap ⚠️

Key Insight: Height ≠ Side Length

Coordinate Geometry Areas

Shaded Region Problems

SAT Trap ⚠️

The Proportion Rule

Inscribed Angle Theorem

Tangent Lines

Surface Area

Common SAT Problem: Filling and Draining

Scaling Rule for 3D

Midpoint Formula

Slope

Equation of a Circle

Converting General Form to Standard Form (Completing the Square)

SAT Trap ⚠️

Common SAT Geometry Question Patterns

Strategy: Draw It

Top 3 Geometry Mistakes