Essential Workbook

Geometry

Core Problems & Concepts

20 exam-style short-answer problems across 5 fundamental units. Study the concept, try the practice problems, then check your solutions.

U1
Lines & Angles
Problems 1 – 4
U2
Triangles
Problems 5 – 8
U3
Pythagorean Theorem
Problems 9 – 12
U4
Quadrilaterals & Polygons
Problems 13 – 16
U5
Circles
Problems 17 – 20
U1 Lines & Angles
Core Concepts
Key Definitions
  • Complementary angles: two angles whose measures sum to 90°.
  • Supplementary angles: two angles whose measures sum to 180°.
  • Vertical angles: opposite angles formed by two intersecting lines — always congruent.
  • Linear pair: adjacent supplementary angles that together form a straight line.
Parallel Lines Cut by a Transversal
  • Corresponding angles are congruent (F-shape).
  • Alternate interior angles are congruent (Z-shape).
  • Co-interior (same-side interior) angles are supplementary.
Key Formulas
a + b = 90°  (complementary)
a + b = 180°  (supplementary)
∠1 = ∠2  (vertical angles)
✦ Worked Example
Q: Two supplementary angles are in the ratio 2 : 3. Find each angle.
Let the angles be 2x and 3x.   2x + 3x = 180  ⟹  x = 36.   Angles: 72° and 108°.
U2 Triangles
Core Concepts
Triangle Angle Sum
  • The sum of interior angles of any triangle = 180°.
  • Exterior angle of a triangle = sum of the two non-adjacent interior angles.
Congruence & Similarity
  • Congruence shortcuts: SSS, SAS, ASA, AAS, HL (right triangles).
  • Similarity shortcuts: AA, SAS~, SSS~.
  • In similar triangles, corresponding sides are proportional and corresponding angles are equal.
Key Formulas
A + B + C = 180°
exterior angle = sum of two non-adjacent interior angles
Area = ½ × base × height
✦ Worked Example
Q: In triangle ABC, angle A = 55° and angle B = 75°. Find angle C.
C = 180 − 55 − 75 = 50°.
U3 Pythagorean Theorem
Core Concepts
The Theorem
  • In a right triangle with legs a, b and hypotenuse c:  a² + b² = c².
  • The hypotenuse is always the side opposite the right angle (the longest side).
Common Pythagorean Triples
  • 3-4-5,   5-12-13,   8-15-17,   7-24-25.
  • Multiples also work: 6-8-10, 9-12-15, etc.
Key Formulas
a² + b² = c²
leg: a = √(c² − b²)
distance: d = √((x₂−x₁)² + (y₂−y₁)²)
✦ Worked Example
Q: A right triangle has legs of length 9 and 12. Find the hypotenuse.
c = √(81 + 144) = √225 = 15.
U4 Quadrilaterals & Polygons
Core Concepts
Key Quadrilateral Properties
  • Parallelogram: opposite sides parallel & equal; opposite angles equal; diagonals bisect each other.
  • Rectangle: parallelogram + 4 right angles; diagonals equal.
  • Rhombus: parallelogram + 4 equal sides; diagonals are perpendicular bisectors.
  • Square: rectangle + rhombus; all properties of both.
  • Trapezoid: exactly one pair of parallel sides (bases).
Interior Angle Sum
  • Sum of interior angles of an n-gon = (n − 2) × 180°.
  • Each interior angle of a regular n-gon = (n − 2) × 180 ÷ n.
Key Formulas
S = (n − 2) × 180°
regular interior angle = (n−2) × 180 / n
exterior angle sum = 360° (any convex polygon)
✦ Worked Example
Q: Find the sum of interior angles of a hexagon.
S = (6 − 2) × 180 = 4 × 180 = 720°.
U5 Circles
Core Concepts
Circle Vocabulary & Angle Rules
  • Radius (r): distance from center to any point on the circle. Diameter d = 2r.
  • Chord: segment with both endpoints on the circle.
  • Central angle = arc it intercepts.
  • Inscribed angle = ½ × intercepted arc.
Tangent & Secant Rules
  • A tangent is perpendicular to the radius at the point of tangency.
  • Two tangent segments from the same external point are equal in length.
Key Formulas
C = 2πr = πd
A = πr²
arc length = (θ/360) × 2πr
sector area = (θ/360) × πr²
✦ Worked Example
Q: A circle has radius 7. Find its circumference and area. (Use π ≈ 3.14)
C = 2 × 3.14 × 7 = 43.96  |  A = 3.14 × 49 = 153.86.
Practice Problems
Write your answer in the box, then click Check. Your score is tracked above.
Answer Key & Solutions
Complete solutions for all 20 problems.