📐 Geometry Mini-Flashcards

Master essential geometric facts and formulas through quick-reference flashcards. These key relationships will help you solve geometry problems efficiently.

🎯 Essential Angle Facts

Parallel Lines

• Corresponding angles: Equal when lines are parallel
• Alternate interior angles: Equal when lines are parallel
• Alternate exterior angles: Equal when lines are parallel
• Same-side interior angles: Supplementary when lines are parallel
• Same-side exterior angles: Supplementary when lines are parallel

Triangle Angles

• Sum of angles: $180°$ in any triangle
• Exterior angle: Equal to sum of opposite interior angles
• Isosceles triangle: Base angles equal
• Equilateral triangle: All angles equal $60°$
• Right triangle: One angle equals $90°$

Circle Angles

• Central angle: Angle with vertex at center
• Inscribed angle: Angle with vertex on circle
• Inscribed angle theorem: Inscribed angle = half central angle
• Thales’ theorem: Angle inscribed in semicircle is right angle
• Cyclic quadrilateral: Opposite angles sum to $180°$

🔢 Essential Length Formulas

Triangle Formulas

• Pythagorean theorem: $a^2 + b^2 = c^2$ (right triangle)
• Area: $A = \frac{1}{2}bh$ or $A = \frac{1}{2}ab\sin C$
• Semi-perimeter: $s = \frac{a + b + c}{2}$
• Heron’s formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$

Special Right Triangles

• 30-60-90: Sides in ratio $1 : \sqrt{3} : 2$
• 45-45-90: Sides in ratio $1 : 1 : \sqrt{2}$
• 3-4-5: Pythagorean triple
• 5-12-13: Pythagorean triple
• 8-15-17: Pythagorean triple

Circle Formulas

• Circumference: $C = 2\pi r = \pi d$
• Area: $A = \pi r^2$
• Arc length: $s = r\theta$ (where $\theta$ is in radians)
• Sector area: $A = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians)

📊 Essential Area Formulas

Basic Shapes

• Rectangle: $A = lw$
• Square: $A = s^2$
• Parallelogram: $A = bh$
• Trapezoid: $A = \frac{1}{2}h(b_1 + b_2)$
• Triangle: $A = \frac{1}{2}bh$

Circle and Ellipse

• Circle: $A = \pi r^2$
• Ellipse: $A = \pi ab$ (where $a$ and $b$ are semi-axes)
• Sector: $A = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians)
• Segment: $A = \frac{1}{2}r^2(\theta - \sin\theta)$ (where $\theta$ is in radians)

Regular Polygons

• Regular $n$-gon: $A = \frac{1}{2}ap$ (where $a$ is apothem, $p$ is perimeter)
• Regular triangle: $A = \frac{\sqrt{3}}{4}s^2$
• Regular square: $A = s^2$
• Regular pentagon: $A = \frac{1}{4}\sqrt{25 + 10\sqrt{5}}s^2$
• Regular hexagon: $A = \frac{3\sqrt{3}}{2}s^2$

⚡ Quick Reference Facts

Angle Relationships

• Complementary angles: Sum to $90°$
• Supplementary angles: Sum to $180°$
• Vertical angles: Equal
• Adjacent angles: Share a common side
• Linear pair: Adjacent and supplementary

Triangle Properties

• Scalene: All sides different lengths
• Isosceles: Two sides equal
• Equilateral: All sides equal
• Right: One angle equals $90°$
• Acute: All angles less than $90°$
• Obtuse: One angle greater than $90°$

Circle Properties

• Radius: Distance from center to edge
• Diameter: Distance across circle through center
• Chord: Line segment connecting two points on circle
• Tangent: Line touching circle at exactly one point
• Secant: Line intersecting circle at two points

🎯 Practice Drills

5-Minute Sprint: Angle Facts

Target: 20 problems in 5 minutes (95%+ accuracy)

1. In triangle ABC, if angle A = 40° and angle B = 60°, what is angle C?
2. If two parallel lines are cut by a transversal, and one angle is 70°, what is its corresponding angle?
3. In a right triangle, if one acute angle is 30°, what is the other acute angle?
4. If an inscribed angle intercepts an arc of 80°, what is the measure of the inscribed angle?
5. In a cyclic quadrilateral, if one angle is 100°, what is the opposite angle?

5-Minute Sprint: Length Formulas

Target: 15 problems in 5 minutes (90%+ accuracy)

1. In a 30-60-90 triangle, if the short leg is 3, what is the hypotenuse?
2. In a 45-45-90 triangle, if one leg is 5, what is the hypotenuse?
3. In a right triangle with legs 3 and 4, what is the hypotenuse?
4. If a circle has radius 7, what is its circumference?
5. If a circle has diameter 10, what is its area?

5-Minute Sprint: Area Formulas

Target: 20 problems in 5 minutes (90%+ accuracy)

1. What is the area of a rectangle with length 8 and width 6?
2. What is the area of a triangle with base 10 and height 8?
3. What is the area of a circle with radius 5?
4. What is the area of a trapezoid with bases 6 and 10 and height 4?
5. What is the area of a parallelogram with base 12 and height 7?

🔍 Advanced Geometric Facts

Similarity

• AA Similarity: Two angles equal
• SAS Similarity: Two sides proportional, included angle equal
• SSS Similarity: All three sides proportional
• Proportional relationships: Use similarity ratios

Power of a Point

• Secant-Secant: $PA \cdot PB = PC \cdot PD$
• Secant-Tangent: $PA \cdot PB = PT^2$
• Tangent-Tangent: $PT_1 = PT_2$
• Chord-Chord: $PA \cdot PB = PC \cdot PD$

Coordinate Geometry

• Distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Midpoint formula: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
• Slope formula: $m = \frac{y_2-y_1}{x_2-x_1}$
• Shoelace formula: For polygon area

📊 Progress Tracking

Accuracy Targets

• Angle facts: 95%+ accuracy
• Length formulas: 90%+ accuracy
• Area formulas: 90%+ accuracy
• Advanced facts: 85%+ accuracy

Speed Targets

• Angle facts: 4 problems per minute
• Length formulas: 3 problems per minute
• Area formulas: 4 problems per minute
• Advanced facts: 2 problems per minute

Weekly Goals

• Week 1: Master basic angle facts and length formulas
• Week 2: Add area formulas, maintain accuracy
• Week 3: Add advanced facts, increase speed
• Week 4: Master all areas, optimize speed

⚡ Quick Reference

Essential Angle Facts:

• Triangle sum: $180°$ in any triangle
• Parallel lines: Corresponding angles equal
• Circle angles: Inscribed angle = half central angle
• Right triangle: One angle equals $90°$

Essential Length Formulas:

• Pythagorean theorem: $a^2 + b^2 = c^2$
• 30-60-90: Sides in ratio $1 : \sqrt{3} : 2$
• 45-45-90: Sides in ratio $1 : 1 : \sqrt{2}$
• Circle circumference: $C = 2\pi r$

Essential Area Formulas:

• Triangle: $A = \frac{1}{2}bh$
• Rectangle: $A = lw$
• Circle: $A = \pi r^2$
• Trapezoid: $A = \frac{1}{2}h(b_1 + b_2)$

Practice Schedule:

• Daily: 10 minutes of geometry facts practice
• Focus areas: Work on your weakest skills
• Progressive difficulty: Increase complexity over time
• Time pressure: Practice under time constraints
• Accuracy first: Speed comes with accuracy

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