Angle Relationships & Formulas

Complementary, supplementary, vertical and explementary angles plus degree-radian and slope conversions.

Geometric Principles

Core mathematical relationships used in secondary school geometry, engineering blueprints, architectural framing, and trigonometry calculations.

Sum = 90°Right Angle

Two angles are complementary when their measures add up to exactly 90 degrees. Each angle is called the complement of the other.

Formula:

∠A + ∠B = 90°

Example: 35° + 55° = 90°

Application: Right-angle triangle calculations & carpentry miters.

Sum = 180°Linear Pair

Two angles are supplementary when their measures add up to 180 degrees. When placed adjacently, they form a straight continuous line.

Formula:

∠A + ∠B = 180°

Example: 110° + 70° = 180°

Application: Roof pitches, parallel line transversal proofs.

Equal PairOpposite

When two straight lines cross, the angles opposite each other at the intersection point are always equal in magnitude.

Theorem:

∠Vertical 1 = ∠Vertical 2

Always congruent (α = β)

Application: Truss bridges, scissor linkages & surveying.

Sum = 360°Circle

Two angles whose sum equals 360 degrees (one full revolution). An angle plus its reflex exterior angle form a conjugate pair.

Formula:

θ + θ_reflex = 360°

Example: 60° + 300° = 360°

Application: Navigation bearings, circular dials, robotics joints.

Degrees to Radians

rad = deg × (π / 180)

180° = 3.14159 rad (π)

Radians to Degrees

deg = rad × (180 / π)

1 rad ≈ 57.2958°

Slope Percentage (%)

Slope% = tan(θ) × 100%

45° = 100% grade (1:1 rise/run)

Pythagorean Triples

a² + b² = c²

3-4-5 Triangle: 36.87° & 53.13°