🦕 Dino Geometry — The Ultimate Geometry Workbook

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Unit 1 — Points, Lines & Angles

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SUPPLEMENTARY

∠A + ∠B = 180°

Two angles that sum to 180°

COMPLEMENTARY

∠A + ∠B = 90°

Two angles that sum to 90°

VERTICAL ANGLES

∠A = ∠C (opposite)

Vertical angles are always equal

LINEAR PAIR

∠A + ∠B = 180°

Adjacent angles on a straight line

⭐ Must Memorize

Vertical angles are congruent (equal)
A straight angle = 180°; a right angle = 90°
Complementary = 90°; Supplementary = 180°
Parallel lines cut by a transversal: alternate interior angles are equal; co-interior angles sum to 180°

📝 Example

Two supplementary angles are in ratio 2:3. Find each angle.

Let angles = 2x and 3x 2x + 3x = 180° 5x = 180° → x = 36° Angles: 72° and 108° ✓

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Unit 2 — Triangles

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ANGLE SUM

A + B + C = 180°

Sum of interior angles

EXTERIOR ANGLE

∠ext = A + B (remote)

Exterior = sum of non-adjacent interior angles

PYTHAGOREAN

a² + b² = c²

Right triangle: c is the hypotenuse

AREA

A = ½ × b × h

Base times height divided by 2

TRIANGLE INEQUALITY

a + b > c

Sum of any 2 sides > third side

SIMILARITY (AA)

2 angles equal → similar

AA, SAS, SSS similarity

⭐ Must Memorize

Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
Equilateral triangle: all angles = 60°
Isosceles: base angles are equal
Congruence: SSS, SAS, ASA, AAS, HL

📝 Example

A right triangle has legs 9 and 12. Find the hypotenuse.

c² = 9² + 12² c² = 81 + 144 = 225 c = √225 = 15 ✓

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Unit 3 — Quadrilaterals & Polygons

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POLYGON INTERIOR SUM

S = (n−2) × 180°

n = number of sides

EACH INTERIOR (REGULAR)

= (n−2)×180° / n

For a regular polygon

EXTERIOR ANGLE SUM

Always = 360°

For any convex polygon

PARALLELOGRAM AREA

A = b × h

Base times perpendicular height

TRAPEZOID AREA

A = ½(b₁+b₂)×h

Average of parallel bases × height

RHOMBUS AREA

A = ½ × d₁ × d₂

Half-product of diagonals

⭐ Must Memorize

Rectangle diagonals are equal and bisect each other
Rhombus diagonals are perpendicular bisectors of each other
Parallelogram: opposite sides equal and parallel, opposite angles equal
Square has ALL properties of rectangle AND rhombus

📝 Example

Find the sum of interior angles of a hexagon (6 sides).

S = (n−2) × 180° S = (6−2) × 180° S = 4 × 180° = 720° ✓

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Unit 4 — Circles

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CIRCUMFERENCE

C = 2πr = πd

r = radius, d = diameter

AREA

A = πr²

r = radius

ARC LENGTH

L = (θ/360°) × 2πr

θ = central angle in degrees

SECTOR AREA

A = (θ/360°) × πr²

Area of a "pie slice"

INSCRIBED ANGLE

Inscribed = ½ × Central

Inscribed angle = half its arc

TANGENT-RADIUS

Tangent ⊥ Radius

Always perpendicular at point of tangency

⭐ Must Memorize

Inscribed angle in a semicircle = 90°
Equal chords are equidistant from center
Tangent segments from external point are equal
Chord-chord angle = ½(arc₁ + arc₂)

📝 Example

A circle has radius 6. Find the arc length of a 120° central angle.

L = (120/360) × 2π(6) L = (1/3) × 12π L = 4π ≈ 12.57 units ✓

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Unit 5 — Similarity & Proportions

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SCALE FACTOR

k = new / original

Ratio of corresponding sides

PERIMETER RATIO

P₁/P₂ = k

Same as scale factor

AREA RATIO

A₁/A₂ = k²

Square of the scale factor

VOLUME RATIO

V₁/V₂ = k³

Cube of the scale factor

GEOMETRIC MEAN

GM = √(a × b)

Used in altitude-on-hypotenuse theorem

MIDSEGMENT

Midseg = ½ × base

Connects midpoints of two sides

⭐ Must Memorize

Similar triangles: AA, SAS~, SSS~
Corresponding sides of similar figures are proportional
Altitude to hypotenuse creates 3 similar triangles
Side Splitter theorem: parallel line divides sides proportionally

📝 Example

Two similar triangles have scale factor 3:4. If the smaller area is 27 cm², find the larger area.

Area ratio = k² = (3/4)² = 9/16 27/A₂ = 9/16 A₂ = 27 × 16/9 = 48 cm² ✓

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Unit 6 — Surface Area & Volume

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CYLINDER VOL

V = πr²h

SA = 2πr² + 2πrh

CONE VOL

V = ⅓πr²h

SA = πr² + πrl (l = slant)

SPHERE VOL

V = (4/3)πr³

SA = 4πr²

PRISM VOL

V = B × h

B = base area, h = height

PYRAMID VOL

V = ⅓ × B × h

B = base area, h = height

RECTANGULAR PRISM

V = l × w × h

SA = 2(lw + lh + wh)

⭐ Must Memorize

Cone and pyramid volume = ⅓ × (cylinder/prism volume)
Slant height (l) of cone: l² = r² + h²
Lateral area of a regular pyramid = ½ × perimeter × slant height
Hemisphere: V = ⅔πr³

📝 Example

Find the volume of a cone with radius 3 and height 4.

V = ⅓ × π × r² × h V = ⅓ × π × 9 × 4 V = 12π ≈ 37.70 units³ ✓

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Unit 7 — Coordinate Geometry

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DISTANCE

d = √((x₂−x₁)²+(y₂−y₁)²)

Distance between two points

MIDPOINT

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Average of coordinates

SLOPE

m = (y₂−y₁)/(x₂−x₁)

Rise over run

PARALLEL LINES

m₁ = m₂

Same slope, different intercepts

PERPENDICULAR

m₁ × m₂ = −1

Slopes are negative reciprocals

CIRCLE EQUATION

(x−h)²+(y−k)²=r²

Center (h,k), radius r

⭐ Must Memorize

Slope-intercept form: y = mx + b
Point-slope form: y − y₁ = m(x − x₁)
Horizontal line: slope = 0; Vertical line: slope = undefined
Standard form of line: Ax + By = C

📝 Example

Find the midpoint of (2, 8) and (6, 4).

M = ((2+6)/2, (8+4)/2) M = (8/2, 12/2) M = (4, 6) ✓

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Unit 8 — Transformations & Proofs

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TRANSLATION

(x,y) → (x+a, y+b)

Slide by (a, b)

REFLECTION OVER x-AXIS

(x,y) → (x, −y)

Flip over x-axis

REFLECTION OVER y-AXIS

(x,y) → (−x, y)

Flip over y-axis

ROTATION 90° CCW

(x,y) → (−y, x)

About the origin

ROTATION 180°

(x,y) → (−x, −y)

About the origin

DILATION

(x,y) → (kx, ky)

Scale factor k from origin

⭐ Must Memorize

Isometry = transformation preserving size and shape (translations, reflections, rotations)
Dilation is NOT an isometry (changes size)
Reflection over y = x: (x, y) → (y, x)
Rotation 270° CCW = Rotation 90° CW: (x,y) → (y, −x)

📝 Example

Point A(3, 5) is rotated 90° counterclockwise about the origin. What are the new coordinates?

Rule: (x, y) → (−y, x) A(3, 5) → (−5, 3) ✓