Unit 1
Angles & Lines
Key Formulas
Supplementary: ∠A + ∠B = 180°
Complementary: ∠A + ∠B = 90°
Vertical angles: ∠A = ∠B (opposite)
Linear pair: adjacent angles on a line = 180°
- Parallel lines cut by a transversal → alternate interior angles are equal.
- Co-interior (same-side) angles are supplementary.
- Corresponding angles are equal.
Example
Two supplementary angles are in ratio 2:3. Find the smaller angle.
→ 2x + 3x = 180 → x = 36 → Smaller = 2(36) = 72°
Unit 2
Triangles
Key Formulas
Sum of interior angles = 180°
Exterior angle = sum of 2 non-adjacent interior angles
Area = ½ × base × height
Pythagorean Theorem: a² + b² = c² (right triangle)
- Isosceles triangle: two equal sides → two equal base angles.
- Equilateral triangle: all angles = 60°, all sides equal.
- Triangle inequality: sum of any two sides > third side.
Example
A right triangle has legs 6 and 8. Find the hypotenuse.
→ c² = 6² + 8² = 36 + 64 = 100 → c = 10
Unit 3
Similarity & Congruence
Key Conditions
Congruence: SSS, SAS, ASA, AAS, HL
Similarity: AA, SAS~, SSS~
Scale factor k → area ratio = k², volume ratio = k³
- Similar triangles have proportional corresponding sides.
- Mid-segment of a triangle = ½ of the parallel base.
Example
Two similar triangles have sides in ratio 3:5. If the smaller has area 27 cm², find the larger's area.
→ Area ratio = (3/5)² = 9/25 → Area = 27 × (25/9) = 75 cm²
Unit 4
Quadrilaterals
Key Formulas
Sum of interior angles = 360°
Rectangle area = l × w
Parallelogram area = base × height
Trapezoid area = ½(b₁ + b₂) × h
Rhombus area = ½d₁d₂
- Parallelogram: opposite sides equal & parallel, diagonals bisect each other.
- Rectangle: all angles 90°, diagonals equal.
- Rhombus: all sides equal, diagonals perpendicular bisectors.
Example
A trapezoid has parallel sides 8 cm and 12 cm, and height 5 cm. Find the area.
→ A = ½(8+12)×5 = ½(20)×5 = 50 cm²
Unit 5
Circles
Key Formulas
Circumference = 2πr = πd
Area = πr²
Arc length = (θ/360°) × 2πr
Sector area = (θ/360°) × πr²
Central angle = inscribed angle × 2
- Inscribed angle theorem: inscribed angle = ½ × central angle.
- Tangent to a circle is perpendicular to the radius at the point of tangency.
- Power of a point: PA × PB = PC × PD (intersecting chords).
Example
A circle has radius 7. Find the area of a sector with central angle 90°.
→ A = (90/360)×π×7² = ¼×49π = 49π/4 ≈ 38.48
Unit 6
Coordinate Geometry
Key Formulas
Distance = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope m = (y₂−y₁)/(x₂−x₁)
Perpendicular slopes: m₁ × m₂ = −1
Circle: (x−h)² + (y−k)² = r²
- Parallel lines have equal slopes.
- The y-intercept is found by setting x = 0.
Example
Find the distance between (1, 2) and (4, 6).
→ d = √[(4−1)²+(6−2)²] = √[9+16] = √25 = 5
Unit 7
Transformations
Rules
Translation (a,b): (x,y) → (x+a, y+b)
Reflection over x-axis: (x,y) → (x,−y)
Reflection over y-axis: (x,y) → (−x,y)
Rotation 90° CCW about origin: (x,y) → (−y,x)
Rotation 180°: (x,y) → (−x,−y)
Dilation by k from origin: (x,y) → (kx,ky)
Example
Point (3, −2) is rotated 90° CCW about the origin. Find the image.
→ (x,y) → (−y, x) = (−(−2), 3) = (2, 3)
Unit 8
3D Solids
Key Formulas
Cube: V = s³, SA = 6s²
Rectangular prism: V = lwh
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Cone: V = ⅓πr²h, Slant = √(r²+h²)
Sphere: V = (4/3)πr³, SA = 4πr²
Pyramid: V = ⅓Bh (B = base area)
Example
Find the volume of a cylinder with radius 3 and height 10.
→ V = π(3²)(10) = 90π ≈ 282.7
Unit 9
Trigonometry (Right Triangle)
SOH-CAH-TOA
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Special angles:
sin30°=½, cos30°=√3/2, tan30°=1/√3
sin45°=√2/2, cos45°=√2/2, tan45°=1
sin60°=√3/2, cos60°=½, tan60°=√3
Example
In a right triangle, the hypotenuse is 10 and one angle is 30°. Find the side opposite to 30°.
→ sin30° = opp/10 → opp = 10 × ½ = 5
Unit 10
Area, Perimeter & Volume Summary
Quick Reference
Square: P = 4s, A = s²
Rectangle: P = 2(l+w), A = lw
Triangle: P = a+b+c, A = ½bh
Circle: C = 2πr, A = πr²
Sector: A = ½r²θ (θ in radians)
Regular polygon: A = ½ × perimeter × apothem
- For composite figures, break into basic shapes and add or subtract areas.
- Perimeter is always linear; area is always squared; volume is always cubed.
Example
A square has perimeter 36 cm. Find its area.
→ s = 36/4 = 9 → A = 9² = 81 cm²
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