Parallel Lines, Perpendicular Lines and Symmetry

Two tests for a pair of sides

A flat figure is made of sides. The next question is how those sides sit against each other. Two tests answer most of it.

Test	Meaning	Everyday example
parallel	never meet, always the same distance apart	the two long edges of a ruler
perpendicular	cross at a right angle	the corner of a page

Parallel lines can be extended forever and never touch. That is the definition. It is also why the gap between them never changes. A gap that shrank would close in the end.

Perpendicular lines do meet. The angle where they meet is exactly $90°$.

$$\text{perpendicular} \;\Rightarrow\; \text{angle} = 90°$$

Why these two tests sort so much

A quadrilateral has four sides. That makes two pairs of opposite sides. Ask whether each pair is parallel, and the whole family splits:

Pairs of parallel sides	Family
$0$	ordinary quadrilateral
$1$	trapezoid
$2$	parallelogram

One question sorts every quadrilateral into three boxes. No measuring needed.

The perpendicular test splits the last box again. A parallelogram with perpendicular sides has four right angles. That makes it a rectangle.

Each test cuts down what is left. Sorting by tests beats sorting by looks.

Sorting by angle size

Angles give a third test. For triangles it is the main one.

Triangle	Its angles
right	one angle of exactly $90°$
acute	all three under $90°$
obtuse	one angle over $90°$

A right triangle is a category of its own. Only one angle of a triangle can be a right angle. The three angles total $180°$, and two right angles would use all of it.

$$90° + 90° = 180° \;\Rightarrow\; \text{no room for a third angle}$$

The two sides at the right angle are perpendicular. So the angle test and the side test ask the same thing here, from two directions.

Lines of symmetry

A line of symmetry is a fold line. Fold the figure along it. The two halves land on each other, with nothing sticking out.

Figure	Lines of symmetry
square	$4$
rectangle	$2$
equilateral triangle	$3$
regular hexagon	$6$
ordinary parallelogram	$0$

A square has two folds through the middles of opposite sides. It has two more along its diagonals. Four in all.

A rectangle has only the first two. Fold a long rectangle along a diagonal and a corner sticks out. So a diagonal is not a fold line for it.

A regular figure has one line of symmetry per side. A regular hexagon has six sides and six folds.

Why a parallelogram has none

A slanted parallelogram looks balanced. It has no line of symmetry at all.

Try the fold. Along a diagonal, the two halves are the same size but the wrong way round. One leans left and the other leans right. They do not land on each other.

Across the middle it fails too. The slant of one half points the opposite way from the other.

The halves match after a turn, not after a fold. Only the fold counts here.

Putting the tests together

To classify a figure, work through the tests in order:

1. Count the sides.
2. Count the pairs of parallel sides.
3. Check for perpendicular sides or right angles.
4. Count the lines of symmetry.

A figure has four sides, two pairs of parallel sides, four right angles, and its sides are not all equal.

Four sides makes it a quadrilateral. Two parallel pairs makes it a parallelogram. Four right angles makes it a rectangle. Unequal sides rules out a square. It has $2$ lines of symmetry, which agrees.

Worked examples

Common mistakes

Practice problems