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Door 8 · Grade 8

📐 Basic Geometry

A point specifies a location, a line extends endlessly in both directions, and a plane is

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Basic Geometrical Concepts

What are points, lines, and planes? A point is a precise location or position in space. It has no size, no width, no length, and no depth. It is represented by a small dot and named with a capital letter. Think of a point like a tiny GPS pin on a digital map.

A line is a straight path of points that extends infinitely in two opposite directions. It has length but no width. We draw arrows on both ends to show it never stops. Think of a perfectly straight, endless highway stretching across the horizon.

A plane is a flat, two-dimensional surface that extends infinitely in all directions. It has length and width but no thickness. Think of a sheet of paper or a flat tabletop that keeps growing forever.

• Point: A dot named \(A\) (written as Point \(A\)).
• Line: A line passing through points \(A\) and \(B\) (written as \(AB\)).
• Plane: A flat surface named by three…

Worked example

Problem: Identify the type of angle for each of the following measurements: \(45°\), \(90°\), and \(135°\).

  1. *Step 1: Check \(45°\).
  2. Since \(45°\) is greater than \(0°\) and less than \(90°\), it is an acute angle.
  3. *Step 2: Check \(90°\).
  4. An angle that measures exactly \(90°\) is a right angle.
  5. *Step 3: Check \(135°\).

Answer: *Step 3: Check \(135°\).

Pairs of Angles: Complementary & Supplementary

What are complementary and supplementary angles? When we look at pairs of angles, we often find special relationships based on what their measures add up to.

Complementary Angles: Two angles are called complementary if the sum of their measures is exactly \(90°\). If you put them side-by-side, they form a perfect right angle (\(L\)-shape). Think of the letter 'C' in Complementary stands for "Corner" (a \(90°\) corner).

• Formula: \(∠ A + ∠ B = 90°\)
• Example: Angles measuring \(40°\) and \(50°\) are complementary because \(40° + 50° = 90°\). We say that \(40°\) is the complement of \(50°\).

Supplementary Angles: Two angles are called supplementary if the sum of their measures is exactly \(180°\). When placed together, they form a flat, straight line. Think of the letter 'S' in Supplementary stands for "Straight line" (\(180°\) line).

• Formula: \(∠ A + ∠ B =…

Worked example

Problem: Find the complement of an angle that measures \(38°\).

  1. *Step 1: Remember that complementary angles add up to \(90°\).
  2. *Step 2: Let the unknown complement angle be \(x\).
  3. *Step 3: Set up the equation: \(x + 38° = 90°\).
  4. *Step 4: Subtract \(38°\) from \(90°\): \(x = 90° - 38° = 52°\).

Answer: *Step 4: Subtract \(38°\) from \(90°\): \(x = 90° - 38° = 52°\).

Pairs of Angles: Adjacent & Vertically Opposite

What are adjacent and vertically opposite angles? Angles can also be paired based on their positions relative to one another when lines cross or meet.

Adjacent Angles: Two angles are adjacent if they are next-door neighbors. To be adjacent, they must meet three strict rules:

• They share a common vertex (corner point).
• They share a common arm (side).
• They do not overlap (their interiors are completely separate).

Think of two adjacent rooms in a house that share a single common wall.

Vertically Opposite Angles: When two straight lines cross each other like an 'X', they create four angles. The angles that sit directly across from each other are called vertically opposite angles.

• Crucial Property: Vertically opposite angles are always equal in measure.
• Think of an open pair of scissors; as you open the handles wider, the blades open wider by the exact same amount.

Worked example

Problem: In the given figure, two lines intersect. If one of the angles formed is \(55°\), find the measure of its vertically opposite angle and the remaining two angles.

  1. *Step 1: Let the given angle be \(∠ 1 = 55°\).
  2. *Step 2: The vertically opposite angle to \(∠ 1\) is \(∠ 3\).
  3. Since vertically opposite angles are equal, \(∠ 3 = 55°\).
  4. *Step 3: The angle adjacent to \(∠ 1\) on a straight line is \(∠ 2\).
  5. They form a linear pair, so they add up to \(180°\).

Answer: They form a linear pair, so they add up to \(180°\).