🦕 Dino Geometry

Rex's Math Adventure — All Geometry Units

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Hi! I'm Rex!
Let's learn geometry together! 🌟

📚 What's Inside

✅ 20 exam-style problems
✅ All geometry units covered
✅ Concepts + Memory tips
✅ Detailed explanations
✅ Timer challenge mode
✅ Print-ready PDF version

📖 Geometry Concepts

Learn before you practice!

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Study these key concepts! 🧠

🔹 Lines, Rays & Segments

A line extends infinitely in both directions. A ray starts at a point and extends one way. A segment has two endpoints.

ComplementaryTwo angles that sum to 90°

SupplementaryTwo angles that sum to 180°

Vertical AnglesOpposite angles formed by intersecting lines — always equal

Linear PairAdjacent angles on a line — sum to 180°

Alternate Interior Angles = Equal (parallel lines)

Corresponding angles are also equal when lines are parallel

📝 Example

Two parallel lines are cut by a transversal. One angle is 65°. Find the alternate interior angle.
Answer: 65° (alternate interior angles are equal).

✓ 65°

🔹 Triangle Basics

The sum of interior angles of any triangle = 180°. The exterior angle equals the sum of the two non-adjacent interior angles.

EquilateralAll 3 sides equal, all angles = 60°

Isosceles2 equal sides → 2 equal base angles

Pythagorean Thma² + b² = c² (right triangles only)

Triangle InequalitySum of any 2 sides > 3rd side

Area = ½ × base × height

Heron's: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2

📝 Example

A right triangle has legs 6 and 8. Find the hypotenuse.
Solution: c = √(6² + 8²) = √(36 + 64) = √100 = 10

✓ 10

🔹 Polygon Angle Sums

Interior angle sum of an n-gon = (n − 2) × 180°. Each interior angle of a regular n-gon = (n − 2) × 180° / n. Exterior angles of any convex polygon sum to 360°.

QuadrilateralInterior sum = 360°

PentagonInterior sum = 540°

HexagonInterior sum = 720°

Diagonal countn(n−3)/2

Parallelogram Area = base × height

Rhombus Area = (d₁ × d₂) / 2 | Trapezoid Area = ½(b₁+b₂)×h

📝 Example

Each interior angle of a regular polygon is 135°. How many sides does it have?
Solution: (n−2)×180/n = 135 → 180n − 360 = 135n → 45n = 360 → n = 8

✓ 8 sides (octagon)

🔹 Circle Essentials

A central angle equals its intercepted arc. An inscribed angle = ½ × intercepted arc. Tangent lines are perpendicular to the radius at the point of tangency.

Circumference2πr

Areaπr²

Arc Length(θ/360)×2πr

Sector Area(θ/360)×πr²

Inscribed Angle = ½ × Arc

Tangent-chord angle = ½ × intercepted arc

📝 Example

A circle has radius 7. Find the area of a sector with central angle 90°.
Solution: (90/360) × π × 7² = ¼ × 49π = 49π/4 ≈ 38.48

✓ 49π/4

🔹 3D Solids

Prisms: V = Bh, SA = 2B + Ph. Cylinders: V = πr²h. Pyramids: V = ⅓Bh. Cones: V = ⅓πr²h. Spheres: V = 4/3 πr³, SA = 4πr².

Rectangular PrismV = l×w×h

CylinderV = πr²h

ConeV = ⅓πr²h

SphereV = (4/3)πr³

📝 Example

A cone has radius 3 and height 4. Find its volume.
Solution: V = ⅓ × π × 9 × 4 = 12π ≈ 37.70

✓ 12π

🔹 Coordinate Geometry

Work on the coordinate plane using algebraic methods to verify geometric properties and solve problems.

Distanced = √[(x₂−x₁)²+(y₂−y₁)²]

MidpointM = ((x₁+x₂)/2, (y₁+y₂)/2)

Slopem = (y₂−y₁)/(x₂−x₁)

Perp. LinesSlopes multiply to −1

📝 Example

Find the distance between (1, 2) and (4, 6).
Solution: d = √[(4−1)² + (6−2)²] = √[9+16] = √25 = 5

✓ 5

🔹 Similarity & Congruence

Triangles are similar (∼) if corresponding angles are equal and sides are proportional. Similar triangles have ratio of areas = (ratio of sides)².

AA Similarity2 pairs of equal angles → similar

SAS Similarity2 sides proportional + included angle equal

SSS SimilarityAll 3 side ratios equal

Area Ratio= (side ratio)²

📝 Example

Two similar triangles have a side ratio of 3:5. What is the ratio of their areas?
Solution: Area ratio = 3² : 5² = 9 : 25

✓ 9 : 25

🔹 Geometric Proofs

Proofs use definitions, postulates, and theorems logically to establish truth. Common proof methods: two-column proof, paragraph proof, coordinate proof.

ReflexiveA segment/angle equals itself

TransitiveIf a=b and b=c, then a=c

CPCTCCorresponding parts of congruent triangles are congruent

HL TheoremRight triangles: hypotenuse + 1 leg → congruent

📝 Example

Two triangles share a side. Both triangles have all three sides equal (SSS). They are congruent. By CPCTC, all corresponding angles and sides are also equal.

✓ SSS → Congruent → CPCTC

⏱ Time

40:00

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Question 1 of 20

Unit 1

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📋 Answer Key & Explanations

Complete solutions for all 20 questions

Dino Geometry — Problem Set

20 Exam-Style Questions • All Geometry Units • Name: ______________ Date: ______________