Lines, Angles, and Triangles
Key Concepts
- Angle types: complementary, supplementary, vertical, adjacent
- Triangle types: equilateral, isosceles, scalene
- Triangle angle sum = 180°
- Exterior angle theorem
Skills Tested
- Solving for unknown angles
- Applying triangle properties
- Using algebra with geometric relationships
Example Question
In a triangle, two angles measure 40° and 65°. What is the third angle? A. 75° B. Answer: 75° (Because 180 - 40 - 65 = 75)
Triangle Properties and Theorems
Key Concepts
- Pythagorean theorem: a^2 + b^2 = c^2
- Special right triangles:
- 45-45-90: sides are x, x, x\sqrt{2}
- 30-60-90: sides are x, x\sqrt{3}, 2x
- Area = \frac{1}{2} \text{base} \times \text{height}
- Triangle Inequality: sum of two sides > third side
Skills Tested
- Finding side lengths
- Solving for unknowns using triangle rules
- Applying special triangle patterns
Example Question
What is the hypotenuse of a right triangle with legs 6 and 8? A. 10 B. Answer: 10 (Use a^2 + b^2 = c^2: 36 + 64 = 100 \Rightarrow c = \sqrt{100} = 10)
Circles
Key Concepts
- Radius, diameter, circumference, area
- Circumference: C = 2\pi r
- Area: A = \pi r^2
- Arcs, sectors, and central angles
- Equations of circles: (x - h)^2 + (y - k)^2 = r^2
Skills Tested
- Calculating arc lengths and areas of sectors
- Working with circle equations
- Finding center and radius from equation
❓ Example Question
What is the area of a circle with diameter 10? A. 25\pi B. Answer: 25\pi (Radius = 5; area = \pi(5)^2)
Polygons
Key Concepts
- Sum of interior angles: (n - 2) \times 180^\circ
- Each angle in regular polygon: \frac{(n - 2) \times 180^\circ}{n}
- Properties of quadrilaterals (parallelograms, rectangles, trapezoids)
Skills Tested
- Finding missing angles
- Identifying shape properties
- Working with side/angle relationships
Example Question
What is the sum of the interior angles of a hexagon? A. 720° B. Answer: 720° (6 sides → (6 - 2) \times 180 = 720^\circ)
Solid Geometry (3D)
Key Concepts
- Volume and surface area formulas:
- Rectangular prism: V = lwh, SA = 2lw + 2lh + 2wh
- Cylinder: V = \pi r^2 h
- Sphere: V = \frac{4}{3}\pi r^3
- Cone: V = \frac{1}{3} \pi r^2 h
- Cross-sections and nets
Skills Tested
- Calculating volume and surface area
- Understanding 3D shape features
- Solving word problems involving shape dimensions
Example Question
What is the volume of a cylinder with radius 3 and height 4? A. 36\pi B. Answer: 36\pi (Volume = \pi r^2 h = \pi(9)(4) = 36\pi)
Coordinate Geometry & Distance
Key Concepts
- Distance formula: \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
- Midpoint formula: \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
- Slope and equations of lines
- Using coordinates to verify geometric relationships
Skills Tested
- Finding lengths, midpoints, and slopes
- Verifying right angles or congruent segments
- Applying geometric properties in the coordinate plane
Example Question
What is the distance between (3, 4) and (6, 8)? A. 5 B. Answer: 5 (Use distance formula: \sqrt{(6 - 3)^2 + (8 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5