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Mom's Basement University

Geometry Review

What You’ll Learn

Section titled “What You’ll Learn”

In this review lesson you’ll revisit the key ideas from the entire Geometry section and practice mixed problems to prepare for the next topics.

The Concept

Section titled “The Concept”

Geometry builds from basic elements to shapes, measurements, and coordinate applications. Key topics include:

  • Points, lines, planes, segments, rays, and angles
  • Angle relationships (adjacent, vertical, complementary, supplementary, parallel lines and transversals)
  • Triangles (classification, Triangle Sum Theorem, Pythagorean Theorem)
  • Quadrilaterals and polygons (properties, interior/exterior angles)
  • Circles (parts, circumference, area, arcs, central angles)
  • Coordinate geometry (distance, midpoint, slope, graphing figures)

Review tips:

  • Draw quick sketches whenever possible.
  • Use the Triangle Sum Theorem (180°) and polygon interior angle formula (n−2)×180°(n - 2) \times 180°.
  • Remember: slope = rise/run, distance uses the Pythagorean Theorem, midpoint averages coordinates.
  • Check units (linear vs. square) and whether answers make sense.

Worked Examples

Section titled “Worked Examples”

1. Angles and Angle Relationships

Two lines intersect. One angle measures 72°. Find the vertical angle and the two adjacent angles.

  • Vertical angle = 72° (vertical angles are equal)
  • Adjacent angles = 180° − 72° = 108° each

2. Triangle Sum Theorem and Classification

In triangle ABC, angle A = 48° and angle B = 67°. Classify the triangle and find angle C.

∠C=180°−48°−67°=65°\angle C = 180° - 48° - 67° = 65°

All angles are less than 90° → acute triangle.

3. Pythagorean Theorem

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

92+122=81+144=225→c=225=15 ft9^2 + 12^2 = 81 + 144 = 225 \quad \rightarrow \quad c = \sqrt{225} = 15 \text{ ft}
a = 6 b = 8 c = 10 6² + 8² = 36 + 64 = 100 = 10²

4. Properties of Quadrilaterals

A quadrilateral has opposite sides parallel and equal, and all angles are 90°. Name it and list two key properties.

  • It is a rectangle (also a parallelogram).
  • Opposite sides equal, diagonals equal, four right angles.

5. Perimeter and Area

A rectangle is 14 m long and 8 m wide.

Perimeter = 2(14 + 8) = 44 m. Area = 14 × 8 = 112 m².

6. Circles: Circumference and Area

A circle has diameter 20 cm (radius = 10 cm).

C=π×20≈62.8 cmA=π×102≈314 cm2\begin{aligned} C &= \pi \times 20 \approx 62.8 \text{ cm} \\[1em] A &= \pi \times 10^2 \approx 314 \text{ cm}^2 \end{aligned}

7. Arcs and Central Angles

A central angle measures 75°.

  • Minor arc = 75° (equals the central angle)
  • Major arc = 360° − 75° = 285°

8. Distance and Midpoint

Find the distance and midpoint between A(−2, 1) and B(4, 5).

d=(4−(−2))2+(5−1)2=36+16=52≈7.21d = \sqrt{(4-(-2))^2 + (5-1)^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21 M=(−2+42,  1+52)=(1,  3)M = \left(\frac{-2+4}{2},\; \frac{1+5}{2}\right) = (1,\; 3)

9. Slope

Find the slope of the line through (1, 3) and (5, 11).

m=11−35−1=84=2m = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2

Positive slope of 2. The line rises 2 units for every 1 unit right.

Real-World Application

Section titled “Real-World Application”

Geometry skills help with everyday and career tasks:

  • Measuring rooms or gardens (area and perimeter).
  • Reading blueprints or maps (angles, slopes, distances).
  • Construction and DIY projects (Pythagorean Theorem for square corners, roof pitches).
  • Design and landscaping (circles, polygons, coordinate planning).
  • Navigation (distances, slopes, graphing routes).

Example: Planning a rectangular patio 12 ft by 8 ft with a circular fountain of radius 3 ft. Perimeter of the patio tells you edging needed, area tells you paving material, and circle area tells you fountain size.

Quiz

Section titled “Quiz”
The sum of interior angles in any triangle is:
What is the slope between points (0,0) and (3,6)?
A circle has radius 6 cm. What is its area?
In a right triangle with legs 5 and 12, the hypotenuse is:
A circle has radius 8 cm. Its area is:
The slope between (0,0) and (3,9) is:
Sum of interior angles in a hexagon is:
Two angles are complementary. One is 38°. The other is:
What is the distance between points $(0, 0)$ and $(6, 8)$?
A triangle has angles $60°$, $60°$, and $60°$. It is classified as:
The midpoint of $(4, 2)$ and $(10, 8)$ is:
A circle has diameter 14 cm. What is its circumference?
In a parallelogram, consecutive angles are:
What is the sum of interior angles of a pentagon?
A right triangle has legs $9$ and $12$. What is the hypotenuse?
Vertical angles are always:
A trapezoid has bases $6$ cm and $10$ cm with height $4$ cm. What is its area?
When a transversal crosses parallel lines, alternate interior angles are:
A central angle of $90°$ creates a minor arc of:
Find the slope between $(-1, 4)$ and $(3, -2)$.