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Door 10 · Grade 8

📐 Quadrilaterals

Any four-sided closed flat shape is a quadrilateral, and its interior angles always sum up

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Properties of Quadrilaterals & Parallelograms

What is a quadrilateral? A quadrilateral is a flat, closed two-dimensional shape made by joining four straight line segments. The word comes from quad meaning four and lateral meaning sides. Real-life examples include a laptop screen, a sticky note, a standard chessboard, or a dollar bill.

Every quadrilateral has:

• Four vertices (corners where two sides meet).
• Four sides (the straight boundary segments).
• Four interior angles (the corners inside the shape).
• Two diagonals (straight lines connecting opposite corners).

One of the most foundational concepts in geometry is the Angle Sum Property of a Quadrilateral. This rule states that the sum of all four interior angles inside any quadrilateral is always exactly \(360°\). If you take any paper quadrilateral and tear off its four corners, you can fit them together perfectly to form a complete circle!

What is a…

Worked example

Problem: Three interior angles of a quadrilateral measure \(65°\), \(115°\), and \(80°\). Find the measure of the fourth interior angle.

  1. *Step 1: Write down the Angle Sum Property formula for a quadrilateral:* \(Properties + ∠ 1 + ∠ 2 + ∠ 3 + ∠ 4 = 360°\)
  2. *Step 2: Substitute the known values into the equation:* \(65° + 115° + 80° + x = 360°\)
  3. *Step 3: Add up the three given numbers:* \(260° + x = 360°\)
  4. *Step 4: Subtract \(260°\) from \(360°\) to calculate the value of \(x\):* \(x = 360° - 260° = 100°\)

Answer: *Step 4: Subtract \(260°\) from \(360°\) to calculate the value of \(x\):* \(x = 360° - 260° = 100°\)

Special Parallelograms: Rectangle, Square, & Rhombus

What are rectangles, squares, and rhombuses? Rectangles, squares, and rhombuses are special members of the parallelogram family. Because they are parallelograms, they automatically inherit all regular parallelogram features (like opposite sides being equal and diagonals bisecting each other). However, each has unique "superpowers" that set them apart.

Rectangle: A rectangle is a parallelogram with four perfect right angles (\(90°\) each). Think of a standard brick, a smartphone screen, or a doorway.

• Special Superpower: Its diagonals are completely equal in length.

Rhombus: A rhombus is a parallelogram where all four sides are exactly equal in length. Think of a diamond pattern on a playing card or a kite shape with equal edges.

• Special Superpower: Its diagonals cross each other at a perfect \(90°\) right angle. We say they are perpendicular bisectors of each…

Worked example

Problem: In a rectangle \(XYZW\), the diagonal \(XZ\) has a length of \(5x - 3 cm\), and the diagonal \(YW\) has a length of \(2x + 6 cm\). Find the value of \(x\) and the actual length of the diagonals.

  1. *Step 1: Recall the unique property of rectangles: the diagonals are completely equal in length.
  2. *Step 2: Set the two expressions equal to each other:* \(5x - 3 = 2x + 6\)
  3. *Step 3: Subtract \(2x\) from both sides of the equation:* \(3x - 3 = 6\)
  4. *Step 4: Add \(3\) to both sides:* \(3x = 9\)
  5. *Step 5: Divide by \(3\):* \(x = 3\)

Answer: *Step 5: Divide by \(3\):* \(x = 3\)

Non-Parallelogram Quadrilaterals: Trapezium & Kite

What are trapeziums and kites? Not all quadrilaterals have two pairs of parallel sides. Some have only one pair, and some have none at all! These are the non-parallelogram members of the quadrilateral family.

Trapezium: A trapezium (often called a trapezoid) is a quadrilateral with exactly one pair of opposite sides that are parallel. The other two sides are non-parallel. Think of a popcorn bucket profile or a handbag shape.

• Isosceles Trapezium: A special type of trapezium where the two non-parallel side legs are exactly equal in length. In this case, the base corner angles are also equal to each other!

Kite: A kite is a quadrilateral that looks exactly like the flying toy you use at the park. It has no parallel sides. Instead, it features two distinct pairs of adjacent sides that are equal in length.

• Special Diagonal Properties: The diagonals of a kite cross…

Worked example

Problem: In a trapezium \(ABCD\), side \(AB\) is parallel to side \(CD\). If \(∠ A = 110°\) and \(∠ B = 85°\), calculate the values of the remaining base angles \(∠ D\) and \(∠ C\).

  1. *Step 1: Identify consecutive interior angles between parallel lines.
  2. Because \(AB CD\), the angles along the transversal legs (\(∠ A\) and \(∠ D\)) must add up to \(180°\).
  3. *Step 2: Calculate \(∠ D\):* \(110° + ∠ D = 180° ∠ D = 180° - 110° = 70°\).
  4. *Step 3: Similarly, the angles along the other leg (\(∠ B\) and \(∠ C\)) must also add up to \(180°\):* \(∠ B + ∠ C = 180°\).
  5. *Step 4: Calculate \(∠ C\):* \(85° + ∠ C = 180° ∠ C = 180° - 85° = 95°\).

Answer: *Step 4: Calculate \(∠ C\):* \(85° + ∠ C = 180° ∠ C = 180° - 85° = 95°\).