Windows, books, and tiles are all quadrilaterals, shapes with four sides.
A parallelogram has opposite sides parallel and equal, opposite angles equal, and diagonals that bisect each other.
Add right angles for a rectangle, equal sides for a rhombus, and both together make a square.
Adjust the shear and side ratio to morph a parallelogram into these shapes.
Intuition — Morphing a Parallelogram
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Tweak shear and ratio to morph parallelogram → square
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👀 How Quadrilaterals Relate
①Parallelogram: opposite sides parallel and equal
②Rectangle: parallelogram with all angles 90°
③Rhombus: parallelogram with all sides equal
④Square: rectangle ∩ rhombus (all properties)
Properties of a Parallelogram
Core Properties
Opposite sides parallel & equal, opposite angles equal, diagonals bisect each other
These three are the basic properties
💡 Tests for Parallelogram
①Opposite sides parallel (definition)
②Opposite sides equal
③Opposite angles equal
④Diagonals bisect each other
⑤One pair of sides parallel and equal
Rectangle, Rhombus, Square
Extra Properties of Rectangle
All angles 90°, diagonals equal
Parallelogram + one 90° angle → rectangle
Extra Properties of Rhombus
All sides equal, diagonals perpendicular
Parallelogram + adjacent sides equal → rhombus
Square
Rectangle ∩ Rhombus = Square
Equal sides, equal 90° angles → has every property
Containment Relations
📝 Containment Summary
①Square ⊂ Rectangle ⊂ Parallelogram
②Square ⊂ Rhombus ⊂ Parallelogram
③Square ⊂ Rectangle ∩ Rhombus
④Parallelogram ⊂ Trapezoid (one parallel pair is enough)
Work It Out
Example 1
In a parallelogram one interior angle is 110°. Find the adjacent interior angle.
1
Adjacent interior angles of a parallelogram sum to 180°.
adjacent angle = 180° − 110°
2
Compute.
= 70°
▸ 70°
In a parallelogram opposite angles are equal and adjacent ones sum to 180°.
Example 2
In parallelogram ABCD, AB = 8 and BC = 5. Find the perimeter.
1
A parallelogram has two pairs of equal opposite sides.
perimeter = 2 × (8 + 5)
2
Compute.
= 26
▸ 26
Opposite sides of a parallelogram are equal in length.
Exam Wrap-up
Core Idea
Parallelogram → Rectangle / Rhombus → Square
Identify the extra condition added at each step
Grade-8 school exam type
In a parallelogram, adjacent angles ∠A and ∠B are in ratio 2 : 3. What is ∠A?
①60°
②72°
③90°
④108°
⑤120°
▸ ② 72°
1
Adjacent angles sum to 180°, so 2k + 3k = 180°.
2k + 3k = 180° ⇒ 5k = 180°
2
k = 36°, so ∠A = 2k.
∠A = 2 × 36° = 72°
🎯 Exam Points
①Proving a parallelogram: any one of the 5 tests is enough
②Rectangle = parallelogram + one 90° (or equal diagonals)