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Dino Geometry
All Topics Β· 20 Exam-Style Problems
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Hi there, math explorer! I'm Rex the Geometry Dino! πŸŽ‰
Let's roar through ALL of Geometry together β€” angles, triangles, circles, volume, and more!
40 minutes Β· 20 multiple-choice questions β€” ready to stomp?
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20
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40min
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100pt
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πŸ“š Topics Covered

πŸ”·Lines & Angles β€” parallel lines, transversals, angle pairs
πŸ”·Triangles β€” congruence, similarity, Pythagorean theorem
πŸ”·Polygons & Quadrilaterals β€” interior/exterior angles, properties
πŸ”·Circles β€” arcs, central & inscribed angles, tangents, chords
πŸ”·Coordinate Geometry β€” distance, midpoint, slope, equations
πŸ”·Transformations β€” translations, reflections, rotations, dilations
πŸ”·Area & Perimeter β€” 2D figures
πŸ”·Surface Area & Volume β€” 3D solids
πŸ”·Trigonometry β€” right-triangle trig ratios
πŸ”·Proofs & Logic β€” conditionals, bi-conditionals
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Before we roar into problems, let me stomp through the KEY concepts and formulas you MUST memorize! πŸ¦–βœοΈ
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Lines & Angles
Parallel lines Β· Transversals Β· Angle relationships

Angle Pair Relationships

When two parallel lines are cut by a transversal:

β€’ Corresponding angles are equal (same position at each intersection)

β€’ Alternate interior angles are equal (between parallel lines, opposite sides)

β€’ Alternate exterior angles are equal (outside parallel lines, opposite sides)

β€’ Co-interior (same-side interior) angles are supplementary (sum = 180Β°)

β€’ Vertical angles are always equal

β€’ Linear pair always sums to 180Β°

⭐ MEMORIZE: Angle Sums

β˜…Supplementary angles: ∠A + ∠B = 180Β°
β˜…Complementary angles: ∠A + ∠B = 90Β°
β˜…Vertical angles: ∠A = ∠C (opposite)
β˜…Co-interior angles: ∠A + ∠B = 180Β°
πŸ“— Quick Example
Two parallel lines are cut by a transversal. One co-interior angle is 65Β°. Find the other.
βœ… Answer: 180Β° βˆ’ 65Β° = 115Β° (co-interior angles are supplementary)
πŸ”Ί
Triangles
Congruence Β· Similarity Β· Pythagorean Theorem Β· Special triangles

Triangle Angle Sum & Exterior Angle Theorem

β€’ Sum of interior angles of any triangle = 180Β°

β€’ An exterior angle of a triangle equals the sum of the two non-adjacent interior angles

β€’ Triangle Inequality: sum of any two sides must be greater than the third side

Congruence (SSS, SAS, ASA, AAS, HL) & Similarity (AA, SAS, SSS)

β€’ Congruent triangles have the same shape AND size (all sides and angles equal)

β€’ Similar triangles have the same shape but different sizes (corresponding angles equal, sides proportional)

β€’ AA Similarity: if two angles of one triangle equal two angles of another, the triangles are similar

⭐ MEMORIZE: Key Formulas

β˜…Pythagorean Theorem: aΒ² + bΒ² = cΒ² (c = hypotenuse)
β˜…45-45-90 triangle: legs = x, hypotenuse = x√2
β˜…30-60-90 triangle: short leg = x, long leg = x√3, hyp = 2x
β˜…Area of triangle: A = Β½ Γ— base Γ— height
β˜…Midsegment of triangle = Β½ Γ— base (parallel to base)
πŸ“— Quick Example
In a 30-60-90 triangle, the short leg is 7. Find the hypotenuse.
βœ… Answer: Hypotenuse = 2 Γ— short leg = 2 Γ— 7 = 14
πŸ”·
Polygons & Quadrilaterals
Interior/exterior angles Β· Properties of special quadrilaterals

⭐ MEMORIZE: Polygon Angle Sums

β˜…Sum of interior angles of n-gon: (n βˆ’ 2) Γ— 180Β°
β˜…Each interior angle of regular n-gon: [(nβˆ’2)Γ—180Β°] Γ· n
β˜…Sum of exterior angles of ANY convex polygon: 360Β°
β˜…Parallelogram: opposite sides β€– and =, opposite angles =, diagonals bisect each other
β˜…Rectangle: parallelogram with 4 right angles, diagonals equal
β˜…Rhombus: parallelogram with 4 equal sides, diagonals perpendicular bisectors of each other
β˜…Square: rectangle + rhombus (all properties of both)
πŸ“— Quick Example
What is the sum of interior angles of a hexagon?
βœ… Answer: (6 βˆ’ 2) Γ— 180Β° = 4 Γ— 180Β° = 720Β°
β­•
Circles
Arcs Β· Central & inscribed angles Β· Tangents Β· Chords

⭐ MEMORIZE: Circle Theorems

β˜…Circumference: C = 2Ο€r = Ο€d
β˜…Area: A = Ο€rΒ²
β˜…Arc length: L = (ΞΈ/360Β°) Γ— 2Ο€r
β˜…Sector area: A = (ΞΈ/360Β°) Γ— Ο€rΒ²
β˜…Central angle = intercepted arc (degree measure)
β˜…Inscribed angle = Β½ Γ— intercepted arc
β˜…Angle formed by two chords: Β½(arc₁ + arcβ‚‚)
β˜…Tangent-radius forms 90Β° angle at point of tangency
β˜…Tangent-chord angle = Β½ Γ— intercepted arc
πŸ“— Quick Example
A circle has radius 6. What is the area of a sector with central angle 90Β°?
βœ… Answer: A = (90/360) Γ— Ο€(6Β²) = ΒΌ Γ— 36Ο€ = 9Ο€
πŸ“Š
Coordinate Geometry
Distance Β· Midpoint Β· Slope Β· Line equations

⭐ MEMORIZE: Coordinate Formulas

β˜…Distance: d = √[(xβ‚‚βˆ’x₁)Β² + (yβ‚‚βˆ’y₁)Β²]
β˜…Midpoint: M = ((x₁+xβ‚‚)/2, (y₁+yβ‚‚)/2)
β˜…Slope: m = (yβ‚‚βˆ’y₁)/(xβ‚‚βˆ’x₁)
β˜…Parallel lines: same slope (m₁ = mβ‚‚)
β˜…Perpendicular lines: slopes are negative reciprocals m₁ Γ— mβ‚‚ = βˆ’1
β˜…Circle equation: (xβˆ’h)Β² + (yβˆ’k)Β² = rΒ²
πŸ“¦
Area, Surface Area & Volume
2D shapes Β· 3D solids

⭐ MEMORIZE: 2D Area Formulas

β˜…Rectangle: A = l Γ— w
β˜…Parallelogram: A = b Γ— h
β˜…Trapezoid: A = Β½(b₁ + bβ‚‚) Γ— h
β˜…Regular polygon: A = Β½ Γ— apothem Γ— perimeter

⭐ MEMORIZE: 3D Volume & Surface Area

β˜…Prism: V = B Γ— h, SA = 2B + lateral area
β˜…Cylinder: V = Ο€rΒ²h, SA = 2Ο€rΒ² + 2Ο€rh
β˜…Cone: V = β…“Ο€rΒ²h, SA = Ο€rΒ² + Ο€rl (l = slant height)
β˜…Pyramid: V = β…“Bh
β˜…Sphere: V = (4/3)Ο€rΒ³, SA = 4Ο€rΒ²
πŸ“
Right Triangle Trigonometry
SOH-CAH-TOA Β· Angle of elevation/depression

⭐ MEMORIZE: Trig Ratios (SOH-CAH-TOA)

β˜…sin ΞΈ = Opposite / Hypotenuse
β˜…cos ΞΈ = Adjacent / Hypotenuse
β˜…tan ΞΈ = Opposite / Adjacent
β˜…Key values: sin30Β°=Β½, cos30Β°=√3/2, tan30Β°=1/√3
β˜…sin60Β°=√3/2, cos60Β°=Β½, tan60Β°=√3
β˜…sin45Β°=cos45Β°=√2/2, tan45Β°=1
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Roar! πŸ¦– Here are your 20 Geometry challenges! Pick the BEST answer for each!
πŸ“ Lines & Angles
1
Two parallel lines are cut by a transversal. One pair of co-interior (same-side interior) angles has one angle measuring 112Β°. What is the measure of the other co-interior angle?
Lines & AnglesEasy
2
Two lines intersect. One of the angles formed is (3x + 20)Β° and its vertical angle is (5x βˆ’ 8)Β°. What is x?
Lines & AnglesMedium
πŸ”Ί Triangles
3
In triangle ABC, angle A = 48Β° and angle B = 73Β°. An exterior angle is drawn at vertex C. What is the measure of this exterior angle at C?
TrianglesEasy
4
In a right triangle, the two legs measure 9 and 12. What is the length of the hypotenuse?
Pythagorean TheoremEasy
5
Triangles PQR and STU are similar with PQ corresponding to ST. If PQ = 8, QR = 12, and ST = 6, what is TU?
Similar TrianglesMedium
πŸ”· Polygons & Quadrilaterals
6
Each interior angle of a regular polygon measures 144Β°. How many sides does the polygon have?
PolygonsMedium
7
In parallelogram ABCD, angle A = (4x + 10)Β° and angle B = (2x + 50)Β°. Since A and B are consecutive angles (co-interior), what is the value of x?
QuadrilateralsMedium
β­• Circles
8
A circle has center O. The central angle AOB = 80Β°. What is the measure of the inscribed angle ACB that intercepts the same arc AB, where C is on the major arc?
CirclesEasy
9
A circle has radius 10 cm. Two chords intersect inside the circle. The chord segments on one chord have lengths 4 and 9. One segment of the other chord has length 6. What is the length of the remaining segment of the second chord?
Circles β€” ChordsMedium
10
A circle has radius 9. What is the arc length of a sector whose central angle is 120Β°? (Leave answer in terms of Ο€.)
Circles β€” Arc LengthMedium
πŸ“Š Coordinate Geometry
11
What is the distance between points A(1, βˆ’2) and B(7, 6)?
Coordinate GeometryEasy
12
Line β„“ has slope 3/4. What is the slope of a line perpendicular to β„“?
Coordinate GeometryEasy
πŸ” Transformations
13
Point P(3, βˆ’5) is reflected over the x-axis. What are the coordinates of its image P'?
TransformationsEasy
14
Triangle XYZ has vertices X(2, 4), Y(6, 4), Z(4, 8). It is dilated by a scale factor of 1/2 centered at the origin. What is the image of vertex Z?
Transformations β€” DilationMedium
πŸ“ Area & Perimeter
15
A trapezoid has parallel bases of 7 cm and 13 cm, and a height of 8 cm. What is its area?
AreaEasy
πŸ“¦ Surface Area & Volume
16
A cylinder has radius 5 cm and height 12 cm. What is its volume? (Leave in terms of Ο€.)
VolumeEasy
17
A cone has a base radius of 6 and a height of 8. What is the total surface area of the cone? (Leave in terms of Ο€.)
Surface AreaHard
Hint: slant height β„“ = √(rΒ² + hΒ²)
πŸ“ Trigonometry
18
In right triangle DEF, angle D = 90Β°, angle E = 30Β°, and EF (hypotenuse) = 20. What is the length of DF (the side opposite to angle E)?
TrigonometryMedium
πŸ” Proofs & Logic
19
Consider the conditional statement: "If a quadrilateral is a rectangle, then it has four right angles." What is the contrapositive of this statement?
Logic & ProofsMedium
🌟 Challenge: Multi-Topic
20
A sphere has a volume of (500/3)Ο€ cmΒ³. What is its surface area? (Leave in terms of Ο€.)
Volume & Surface AreaHard
Hint: V = (4/3)Ο€rΒ³  |  SA = 4Ο€rΒ²
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πŸ“– Answer Key & Explanations
Full worked solutions by Rex the Geometry Dino πŸ¦•