📐 Core Concepts & Formulas
Angles & Lines
Complementary · Supplementary · Transversals
Complementary Angles
Two angles that sum to 90°
∠A + ∠B = 90°
∠A + ∠B = 90°
Supplementary Angles
Two angles that sum to 180°
∠A + ∠B = 180°
∠A + ∠B = 180°
Vertical Angles
Opposite angles formed by two intersecting lines — always equal
Parallel Lines + Transversal
Corresponding ∠s = equal
Alternate interior ∠s = equal
Co-interior ∠s = 180°
Alternate interior ∠s = equal
Co-interior ∠s = 180°
🧠 Must Memorize
- Vertical angles are always congruent (equal)
- A straight line = 180° (linear pair)
- Angles in a full revolution = 360°
- Alternate interior angles are equal when lines are parallel
✏ Worked Example
Two lines intersect. One angle is 65°. Find the vertically opposite angle and its supplement.
Vertical angle = 65° | Supplement = 180° − 65° = 115°
Triangles
Interior angles · Similarity · Congruence · Pythagorean Theorem
Angle Sum
∠A + ∠B + ∠C = 180°
Pythagorean Theorem
a² + b² = c²
(right triangle only)
(right triangle only)
Area
A = ½ × base × height
Exterior Angle
Exterior ∠ = sum of the two non-adjacent interior angles
🧠 Must Memorize
- Pythagorean triples: (3,4,5) · (5,12,13) · (8,15,17)
- Equilateral triangle: all sides equal, each angle = 60°
- Isosceles triangle: base angles are equal
- Exterior angle = sum of two remote interior angles
✏ Worked Example
A right triangle has legs of 6 and 8. Find the hypotenuse.
c = √(6² + 8²) = √(36 + 64) = √100 = 10
Polygons & Quadrilaterals
Interior angles · Perimeter · Area · Properties
Sum of Interior Angles
S = (n − 2) × 180°
(n = number of sides)
(n = number of sides)
Each Interior Angle (regular)
= (n − 2) × 180° ÷ n
Rectangle Area
A = length × width
Trapezoid Area
A = ½ × (a + b) × h
a, b = parallel sides
a, b = parallel sides
🧠 Must Memorize
- Sum of exterior angles of any convex polygon = 360°
- Parallelogram: opposite sides parallel & equal; opposite angles equal
- Rhombus: all sides equal; diagonals bisect at right angles
- Square: all sides equal; all angles 90°; diagonals equal & perpendicular
✏ Worked Example
Find the sum of interior angles of a hexagon (6 sides).
S = (6 − 2) × 180° = 4 × 180° = 720°
Circles
Circumference · Area · Arc · Sector · Chord theorems
Circumference
C = 2πr = πd
Area
A = πr²
Arc Length
L = (θ/360°) × 2πr
Sector Area
A = (θ/360°) × πr²
🧠 Must Memorize
- Inscribed angle = ½ × central angle (same arc)
- Angle in a semicircle = 90°
- Tangent meets radius at 90°
- Equal chords are equidistant from the center
✏ Worked Example
A circle has radius 7 cm. Find its area and circumference. (Use π ≈ 3.14)
Area = π × 7² = 49π ≈ 153.94 cm² | C = 2π × 7 = 14π ≈ 43.96 cm
Coordinate Geometry
Distance · Midpoint · Slope · Equations
Distance Formula
d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope
m = (y₂−y₁) / (x₂−x₁)
Perpendicular Slopes
m₁ × m₂ = −1
(negative reciprocals)
(negative reciprocals)
🧠 Must Memorize
- Parallel lines have equal slopes
- Perpendicular slopes are negative reciprocals
- Slope-intercept form: y = mx + b
- Horizontal line: slope = 0 | Vertical line: undefined slope
✏ Worked Example
Find the distance between A(1, 2) and B(4, 6).
d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5
📝 Practice Problems
20 Core Geometry Problems
Select the best answer for each question. Explanations appear immediately after each answer.
0
out of 20
Great Job!
Keep practicing to reach mastery.
0
Correct
0
Incorrect
—
Time Used