📚 Key Concepts — Memorize These

Geometry Core Formulas & Theorems

Triangle Angles
∠A + ∠B + ∠C = 180°
Exterior ∠ = sum of non-adjacent interior ∠s
Pythagorean Theorem
a² + b² = c²
(c = hypotenuse)
Circle
Area = πr²
Circumference = 2πr
Arc length = (θ/360)·2πr
Sector area = (θ/360)·πr²
Polygon Interior Angles
Sum = (n−2)×180°
Each angle (regular) = (n−2)×180°/n
Volume
Cylinder: πr²h
Cone: ⅓πr²h
Sphere: (4/3)πr³
Surface Area
Rectangular prism:
2(lw + lh + wh)
Coordinate Geometry
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Distance: √[(x₂−x₁)²+(y₂−y₁)²]
Circle: (x−h)²+(y−k)²=r²
Lines & Slopes
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1
Similar & Congruent
Similar: AA, SAS~, SSS~
Congruent: SAS, ASA, SSS, AAS
Sides proportional if similar
Inscribed Angle
Inscribed angle = ½ × intercepted arc
Central angle = intercepted arc

✏️ Worked Example

Q: In a triangle, two angles measure 55° and 78°. What is the third angle?
Sum of angles = 180°
Third angle = 180° − 55° − 78° = 47°
Practice Problems — Answer All 20 Questions
1
Triangle Angles
In triangle ABC, ∠A = 52° and ∠B = 73°. What is the measure of ∠C?
Easy
2
Exterior Angle Theorem
An exterior angle of a triangle measures 112°. One of the non-adjacent interior angles is 48°. What is the other non-adjacent interior angle?
Easy
3
Pythagorean Theorem
A right triangle has legs of length 8 and 15. What is the length of the hypotenuse?
Easy
4
Area of a Circle
A circle has a diameter of 10. What is its area? (Use π ≈ 3.14)
Easy
5
Volume of a Cylinder
A cylinder has radius 4 and height 9. What is its volume? (Use π ≈ 3.14)
Medium
6
Similar Triangles
Two similar triangles have corresponding sides in the ratio 3:5. If the shorter triangle has a side of 9, what is the corresponding side of the larger triangle?
Medium
7
Parallel Lines & Transversals
Two parallel lines are cut by a transversal. One alternate interior angle measures 63°. What is the measure of the other alternate interior angle?
Easy
8
Properties of a Parallelogram
In parallelogram PQRS, ∠P = 68°. What is the measure of ∠Q?
Easy
9
Arc Length
A circle has radius 6. What is the arc length of a sector with a central angle of 90°? (Use π ≈ 3.14)
Medium
10
Surface Area — Rectangular Prism
A rectangular prism has length 5, width 3, and height 4. What is its total surface area?
Medium
11
Midpoint Formula
Points A(2, 6) and B(8, 2) are endpoints of a segment. What is the midpoint of AB?
Easy
12
Distance Formula
Find the distance between points P(1, 2) and Q(4, 6).
Medium
13
Perpendicular Slopes
Line ℓ has slope 3/4. What is the slope of a line perpendicular to ℓ?
Medium
14
Interior Angles of a Polygon
What is the sum of interior angles of a regular hexagon?
Medium
15
Sector Area
A circle has radius 6. What is the area of a sector with a central angle of 60°? (Use π ≈ 3.14)
Medium
16
Triangle Congruence — SAS
Two triangles have two pairs of equal sides and the included angles are equal. Which congruence rule applies?
Easy
17
Inscribed Angle Theorem
An inscribed angle in a circle intercepts an arc of 140°. What is the measure of the inscribed angle?
Medium
18
Volume of a Cone
A cone has radius 3 and height 7. What is its volume? (Use π ≈ 3.14)
Medium
19
Equation of a Circle
A circle has center (3, −2) and radius 5. Which equation represents this circle?
Hard
20
Diagonal of a Rectangle
A rectangle has length 12 and width 5. What is the length of its diagonal?
Medium
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