Quadrilaterals (II): Parallelogram & rhombus

1Summary 重點摘要

Parallelogram: opposite sides parallel; rhombus: all sides equal.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Two features of a parallelogramA parallelogram has two main features. It looks like a rectangle pushed sideways: the top and bottom sides are parallel, and the left and right sides are parallel too, so both pairs of opposite sides are parallel. Also, opposite sides are equal: top equals bottom, left equals right, for example top 6 cm and bottom 6 cm, left 3 cm and right 3 cm. Both features together make a parallelogram. A common mistake is to check only one pair of sides; both pairs must be checked. A shape with just one pair (such as a trapezium) is not a parallelogram, and neither condition may be dropped.
2. Part 2: Angles of a parallelogramThe angles of a parallelogram need not be right angles. Two corners may be sharp and two wide, for example 100° and 80° taking turns, but the four angles always add up to 360°: 100° + 80° + 100° + 80° = 360°. Two neighbouring angles add to exactly 180°, for example 100° + 80° = 180°. A common mistake is to think all four angles are equal, or that its angles can never be right angles; rectangles and squares are in fact parallelograms too, with all four angles right angles. The rule is 'need not be right angles', not 'can never be right angles'.
3. Part 3: A rhombus is a special parallelogramA rhombus is a special parallelogram. Besides having opposite sides parallel, all four of its sides are equal in length. Measure with a ruler: if all four sides are 5 cm, it is a rhombus; if even one side differs, for example three sides of 5 cm and one of 4 cm, it is not. A rhombus looks like a diamond or a kite, and its angles, like any parallelogram's, need not be right angles. A common mistake is to count the sides without measuring their lengths and call an ordinary parallelogram a rhombus. To decide, measure every side: only when all four lengths are exactly the same may you conclude rhombus.
4. Part 4: Rhombus vs squareA rhombus differs from a square: a square needs four equal sides and four right angles, while a rhombus has four equal sides but its angles need not be 90°. So every square is a rhombus, but not every rhombus is a square. For example, a rhombus with 5 cm sides and one angle of 70° is not a square. A common mistake is to call any shape with four equal sides a square and forget the right-angle condition, or to think rhombus and square have nothing to do with each other. Think of it as sets: all squares are rhombuses, and among rhombuses only those with right angles are squares. In answers, state the two conditions separately.
5. Part 5: How to checkCheck with a ruler. Line the ruler up with the top side and note its direction, then slide it to the bottom side: do they point the same way? Check the left and right sides the same way. If both pairs are parallel, it is a parallelogram. Measuring works too: top 6 cm and bottom 6 cm, left 3 cm and right 3 cm. To test a rhombus, add one more step: measure all four sides one by one; only if all are equal is it a rhombus. A common mistake is to tilt the ruler while sliding, or to measure only two neighbouring sides. Keep the ruler's direction unchanged at every move, and for a rhombus no side may be skipped.
6. Part 6: Everyday examplesLife is full of examples. Rhombuses include kites, the gaps in wire fences, and floor tile patterns; parallelograms include expanding gates, tilted signs, some stair side panels and collapsible clothes racks. Watch closely: when an expanding gate opens, every gap becomes a parallelogram, because the gate is a set of rectangular frames pushed sideways. A common mistake is to call a kite a parallelogram; most kites are rhombuses with four equal sides. Next time you see these things, count the sides, measure them, then look at the angles and name the shape; geometry then lives in daily life and sticks in memory.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Parallelogram: both pairs of opposite sides parallel and equal
• Rhombus: a parallelogram with all four sides equal
• A rhombus need not have right angles
• A square is a special rhombus (with right angles added)
• Check with a ruler: parallel by sliding, equal by measuring