Dino Geometry - Fun Math Adventure | Geometry Practice Book

Unit 1

Points, Lines & Planes

The building blocks of all geometry!

📍 Core Definitions

Point

Location, no size

Line

∞ in 2 directions

Ray

1 endpoint, ∞ in 1 dir

Plane

Flat surface, ∞ extent

⭐ Must Memorize

• 2 points → exactly 1 line
• 3 noncollinear points → exactly 1 plane
• Collinear = points on the same line
• Coplanar = points on the same plane

Unit 2

Angles & Angle Pairs

From acute to reflex — know them all!

📐 Angle Types

Acute

0° < θ < 90°

Right

θ = 90°

Obtuse

90° < θ < 180°

Straight

θ = 180°

🧮 Key Angle Relationships

Complementary: A + B = 90°
Supplementary: A + B = 180°
Vertical angles are equal

EXAMPLE

Two angles are supplementary. One angle is 3 times the other. Find both angles.

x + 3x = 180° → 4x = 180° → x = 45°, 3x = 135°

Unit 3

Parallel Lines & Transversals

Master the 8 angle positions!

↔️ Angle Relationships

⭐ When lines are parallel (‖):

• Corresponding angles: EQUAL (same side, same position) ∠1=∠5
• Alternate Interior: EQUAL (between lines, opposite sides) ∠3=∠6
• Alternate Exterior: EQUAL (outside lines, opposite sides) ∠1=∠8
• Co-interior (Same-side interior): SUPPLEMENTARY = 180° ∠3+∠5=180°

EXAMPLE

Parallel lines cut by transversal. One co-interior angle = 112°. Find the other.

180° − 112° = 68° (co-interior angles are supplementary)

Unit 4

Triangles

The most important polygon in geometry!

△ Triangle Essentials

🧮 Core Formulas

Angle Sum: A + B + C = 180°
Area: A = ½b×h
Exterior Angle = sum of 2 non-adjacent interior angles

⭐ Triangle Types — Remember!

Equilateral

All 60°, 3 equal sides

Isosceles

2 equal sides & angles

Scalene

No equal sides

Right

One 90° angle

EXAMPLE

A triangle has angles of 50° and 70°. What is the exterior angle at the third vertex?

Third angle = 180° − 50° − 70° = 60°. Exterior = 180° − 60° = 120°.
Or: 50° + 70° = 120° ✓

Unit 5

Pythagorean Theorem

The legendary formula for right triangles!

📏 Right Triangle Magic

🧮 The Formula

a² + b² = c²

c = hypotenuse (longest side, opposite 90°)

⭐ Common Pythagorean Triples

• 3-4-5 (and multiples: 6-8-10, 9-12-15…)
• 5-12-13
• 8-15-17
• 7-24-25

⭐ Special Right Triangles

• 45°-45°-90°: sides = x : x : x√2
• 30°-60°-90°: sides = x : x√3 : 2x

EXAMPLE

A right triangle has legs of 5 and 12. Find the hypotenuse.

c = √(5² + 12²) = √(25 + 144) = √169 = 13

Unit 6

Polygons & Quadrilaterals

From squares to complex shapes!

⬡ Polygon Formulas

🧮 Sum of Interior Angles

S = (n − 2) × 180°

n = number of sides

🧮 Each Interior Angle (regular polygon)

θ = (n−2)×180° ÷ n

⭐ Exterior Angles

Sum of exterior angles of ANY polygon = 360°
Each exterior angle (regular) = 360° ÷ n

Triangle

180°

Quadrilateral

360°

Pentagon

540°

Hexagon

720°

Unit 7

Circles

Round and round we go with π!

⭕ Circle Formulas

🧮 Essential Formulas

Circumference: C = 2πr = πd
Area: A = πr²
Arc Length: L = (θ/360°) × 2πr
Sector Area: A = (θ/360°) × πr²

⭐ Circle Angle Theorems

• Central angle = intercepted arc
• Inscribed angle = ½ × intercepted arc
• Tangent-chord angle = ½ × intercepted arc
• Two chords: angle = ½(arc₁ + arc₂)
• Secant-secant (outside): angle = ½(far arc − near arc)

⭐ Chord & Tangent Theorems

• Tangent⊥radius at point of tangency
• Two tangents from same external point: equal length
• Chord-chord: a×b = c×d
• Secant-secant (external): whole × ext = whole × ext

EXAMPLE

A circle has radius 6 cm. Find the area of a sector with central angle 90°.

A = (90/360) × π × 6² = (1/4) × 36π = 9π ≈ 28.27 cm²

Unit 8

Coordinate Geometry

Geometry meets algebra on the grid!

📊 The Coordinate Toolkit

🧮 Key Formulas

Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂−y₁)/(x₂−x₁)
Circle (center h,k): (x−h)² + (y−k)² = r²

⭐ Slope Relationships

• Parallel lines: same slope (m₁ = m₂)
• Perpendicular lines: m₁ × m₂ = −1 (negative reciprocals)

EXAMPLE

Find the distance between (1, 2) and (4, 6).

d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5

Unit 9

Area & Volume

2D and 3D measurements!

📦 Area Formulas

Rectangle

A = lw

Triangle

A = ½bh

Parallelogram

A = bh

Trapezoid

A = ½(b₁+b₂)h

Circle

A = πr²

Square

A = s²

🧮 Volume Formulas

Rectangular Prism: V = lwh
Cylinder: V = πr²h
Cone: V = ⅓πr²h
Sphere: V = (4/3)πr³
Pyramid: V = ⅓Bh (B = base area)

Unit 10

Transformations, Congruence & Similarity

Moving, matching, and scaling shapes!

🔄 Transformations

⭐ Types of Transformations

• Translation: slide (x+a, y+b) — preserves size & shape
• Reflection: flip over a line
• Rotation: turn around a point
• Dilation: scale by factor k — changes size, preserves shape

🧮 Similarity & Congruence

Similar (~): same shape, different size
Congruent (≅): same shape AND size
Scale factor k: ratios of corresponding sides
Area ratio = k² | Volume ratio = k³

⭐ Triangle Congruence (SSS, SAS, ASA, AAS, HL)

Triangle Similarity (AA, SAS~, SSS~)