Lesson 5 • 50 min read
Geometry & Trigonometry
In This Lesson
1Triangle Similarity
Two triangles are similar if they have the same shape but not necessarily the same size. This means corresponding angles are equal and corresponding sides are proportional.
Scale Factor
a₁/a₂ = b₁/b₂ = c₁/c₂ = k
where k is the scale factor
Similarity Theorems
AA (Angle-Angle)
Two pairs of corresponding angles are congruent
(Third angle is automatically equal)
SSS (Side-Side-Side)
All three pairs of corresponding sides are proportional
a₁/a₂ = b₁/b₂ = c₁/c₂
SAS (Side-Angle-Side)
Two pairs of proportional sides with included angle congruent
a₁/a₂ = b₁/b₂ and ∠B₁ = ∠B₂
Example: Finding Missing Sides
Triangle ABC ~ Triangle DEF with AB=6, BC=8, AC=10 and DE=9
Scale factor k = DE/AB = 9/6 = 1.5
EF = BC × 1.5 = 8 × 1.5 = 12
DF = AC × 1.5 = 10 × 1.5 = 15
2SOH-CAH-TOA (Trigonometric Ratios)
For a right triangle, the trigonometric ratios relate the angles to the sides. SOH-CAH-TOA is a mnemonic to remember them.
SOH
sin θ = O/H
Opposite / Hypotenuse
CAH
cos θ = A/H
Adjacent / Hypotenuse
TOA
tan θ = O/A
Opposite / Adjacent
Identifying Sides
Hypotenuse: Longest side, opposite the right angle
Opposite: Side across from the angle θ
Adjacent: Side next to angle θ (not the hypotenuse)
Example: Find x (opposite side)
Right triangle with angle 30°, hypotenuse = 10
sin 30° = x/10
0.5 = x/10
x = 5
Example: Find angle θ
Right triangle with opposite = 3, adjacent = 4
tan θ = 3/4 = 0.75
θ = tan⁻¹(0.75)
θ ≈ 36.87°
3Special Right Triangles
45-45-90 Triangle
x : x : x√2
legs : legs : hypotenuse
sin 45° = √2/2 ≈ 0.707
cos 45° = √2/2 ≈ 0.707
tan 45° = 1
Isosceles right triangle
30-60-90 Triangle
x : x√3 : 2x
short : long : hypotenuse
sin 30° = 1/2, sin 60° = √3/2
cos 30° = √3/2, cos 60° = 1/2
tan 30° = √3/3, tan 60° = √3
Half of an equilateral triangle
Example: 45-45-90 with leg = 5
Sides: 5 : 5 : 5√2
Hypotenuse = 5√2 ≈ 7.07
Example: 30-60-90 with short leg = 4
Sides: 4 : 4√3 : 8
Long leg = 4√3 ≈ 6.93, Hypotenuse = 8
4Parallelograms
Properties of All Parallelograms
- • Opposite sides are parallel and congruent
- • Opposite angles are congruent
- • Consecutive angles are supplementary (add to 180°)
- • Diagonals bisect each other
Special Parallelograms
Rectangle
- • All angles = 90°
- • Diagonals are congruent
- • Has all parallelogram properties
Rhombus
- • All sides congruent
- • Diagonals are perpendicular
- • Diagonals bisect angles
Square
- • All sides congruent
- • All angles = 90°
- • Both rectangle & rhombus
Hierarchy
Square → Rectangle → Parallelogram
Square → Rhombus → Parallelogram
(A square has ALL properties of rectangles AND rhombi)
5Area Formulas
Parallelogram
A = base × height
A = bh
Rectangle
A = length × width
A = lw
Rhombus
A = (d₁ × d₂) / 2
Half product of diagonals
Square
A = s²
Side squared
Triangle
A = (base × height) / 2
A = ½bh
Trapezoid
A = ½(b₁ + b₂)h
Average of bases × height
Example: Rhombus with diagonals 6 and 8
A = (d₁ × d₂) / 2
A = (6 × 8) / 2 = 48 / 2
A = 24 square units