Chapter 13 · Two-dimensional figures

Polygons and Their Parallel and Perpendicular Sides

Learn what makes a figure a polygon and how to name it by counting sides. Then find parallel and perpendicular sides, even in a figure that has been turned.

13 min read

Triangles and squares are both polygons. A polygon is a closed figure made of straight sides. Both parts of that meaning matter.

  • Closed means the sides join end to end all the way around. If you trace the figure with your finger, you come back to where you started. There is no gap.
  • Straight sides means every side is a line segment, a straight piece of a line with two ends. No part of the figure curves.

A circle is closed, but its edge is curved, so a circle is not a polygon. When you write a capital Z, you draw three straight lines, but the two ends never join. A Z is open, so it is not a polygon either.

Each corner of a polygon is a point where two sides meet. It takes at least 33 straight sides to close a figure. A polygon gets its name from how many sides it has.

Count the sides to find the name of a polygon.
Number of sidesName
33
triangle
44
quadrilateral
55
pentagon
66
hexagon
77
heptagon
88
octagon
Coordinate grid from x 0 to 8 and y 0 to 7, with a grid step of 1, showing 1 plotted element. Details: polygon with 5 vertices.
Drawing A joins (2,1)(2, 1), (6,1)(6, 1), (7,4)(7, 4), (4,6)(4, 6) and (1,4)(1, 4) in order, then goes back to (2,1)(2, 1).

Worked example

Polygon or not?

Picture two drawings on a coordinate plane. In every figure in this lesson, the xx-axis runs across and the yy-axis runs up. Both drawings start at (2,1)(2, 1) and join (6,1)(6, 1), (7,4)(7, 4), (4,6)(4, 6) and (1,4)(1, 4) in order with straight lines. Drawing A then adds one more straight line, from (1,4)(1, 4) back to (2,1)(2, 1). Drawing B stops at (1,4)(1, 4). The figure shows drawing A only. Which drawing is a polygon, and what is its name?

  1. Test drawing B. Is it closed? It starts at (2, 1) and stops at (1, 4). Those are two different points, so the drawing leaves a gap between its end and its start. Drawing B is open, so it is not a polygon.

  2. Test drawing A. Is it closed? Its last line goes from (1, 4) back to the start at (2, 1). Trace around the figure and you return to where you began, so there is no gap. Are its sides straight? Each line goes straight from one point to the next. Both tests pass, so drawing A is a polygon.

  3. Count the sides of drawing A. Joining the five points in order makes 4 sides: (2, 1) to (6, 1), (6, 1) to (7, 4), (7, 4) to (4, 6), and (4, 6) to (1, 4). The line back to the start, from (1, 4) to (2, 1), adds 1 more side.

    4+1=54 + 1 = 5

  4. Match the number of sides to the table. A polygon with 5 sides is a pentagon.

Answer

Drawing A is a polygon with 55 sides, so it is a pentagon. Drawing B is open, so it is not a polygon.

Naming a polygon is a start. You can say more about it by looking at its sides two at a time.

Two sides are parallel when they stay the same distance apart along their whole length. Each side is a piece of a straight line. Picture making two parallel sides longer and longer in both directions. They would never meet. The two rails of a straight train track are a good picture of this.

Two sides are perpendicular when they meet at a right angle. A right angle is a square corner, like the corner of a sheet of paper. It measures 90∘90^\circ.

Many pairs of sides are neither parallel nor perpendicular.

Key concept

Describe a polygon by its sides

A polygon is a closed figure made of straight sides. Count the sides to name it. Then compare the sides two at a time:

  • Parallel sides stay the same distance apart, so they would never meet, however long you made them.
  • Perpendicular sides meet at a right angle.

These are facts about how two sides sit with each other. They stay true however the figure is turned.

On a grid, you can test a pair of sides in two ways.

  • Gap test for parallel sides. Measure the gap between the two sides along grid lines in two different places. Measure the same way both times: both straight up, or both straight across. If the gap is the same, the sides are not getting any closer, so they are parallel. If the gap changes, the sides are getting closer together, so their lines would meet. They are not parallel.
  • Corner test for perpendicular sides. Grid lines cross at square corners. So when a side that runs straight across meets a side that runs straight up, the corner is a right angle. For a corner with slanted sides, fit the corner of a sheet of paper into it. If both sides run along the edges of the paper, the sides are perpendicular.
Coordinate grid from x 0 to 8 and y 0 to 10, with a grid step of 1, showing 2 plotted elements. Details: polygon with 3 vertices; polygon with 4 vertices.
The polygon has corners (1,1)(1, 1), (7,1)(7, 1), (4,5)(4, 5) and (1,5)(1, 5). The helper triangle above it has its top corner at (1,9)(1, 9).

Worked example

Find parallel and perpendicular sides

The figure shows a polygon with corners (1,1)(1, 1), (7,1)(7, 1), (4,5)(4, 5) and (1,5)(1, 5). A helper triangle sits above it, with its top corner at (1,9)(1, 9). You will use the helper in the last step. Name the polygon. Then find its parallel sides and its perpendicular sides.

  1. Count the sides and give each one a name. Bottom side: (1, 1) to (7, 1). Slanted side: (7, 1) to (4, 5). Top side: (4, 5) to (1, 5). Left side: (1, 5) to (1, 1). There are 4 sides, so the polygon is a quadrilateral.

  2. Use the gap test on the bottom and top sides. Measure straight up in two places. From (1, 1) up to (1, 5), the y-coordinate goes from 1 to 5, which is 4 units. From (4, 1) up to (4, 5) is also 4 units. The gap stays the same, so the bottom and top sides are parallel.

    5−1=4and5−1=45 - 1 = 4 \quad \text{and} \quad 5 - 1 = 4

  3. Use the corner test. The left side runs straight up. The bottom and top sides run straight across. So the left side meets the bottom side at a right angle at (1, 1), and it meets the top side at a right angle at (1, 5). Those two pairs are perpendicular. The slanted side does not run straight up, so its two corners are not square corners.

  4. The slanted side meets the bottom and top sides, so it is not parallel to them. Test it against the left side. Measure straight across in two places. Along the bottom side, from (1, 1) to (7, 1), the x-coordinate goes from 1 to 7, which is 6 units. Along the top side, from (1, 5) to (4, 5) is 3 units. The gap shrinks, so these sides are getting closer together.

    7−1=6but4−1=37 - 1 = 6 \quad \text{but} \quad 4 - 1 = 3

  5. Look at the helper triangle to see where that leads. It continues both sides upward. The slanted side goes left 3 and up 4 to reach (4, 5). Go left 3 and up 4 again and you land on (1, 9), straight above the left side. The two lines meet there, so the left side and the slanted side are not parallel, even though they do not touch inside the polygon.

Answer

The polygon is a quadrilateral. Its bottom and top sides are parallel. Its left side is perpendicular to the bottom side and to the top side. The slanted side is not parallel or perpendicular to any other side.

Coordinate grid from x 0 to 8 and y 0 to 8, with a grid step of 1, showing 1 plotted element. Details: polygon with 4 vertices.
The corners are (3,1)(3, 1), (7,3)(7, 3), (5,7)(5, 7) and (1,5)(1, 5). No side runs straight across or straight up.

Worked example

Explain why slanted sides are parallel

The figure shows a quadrilateral with corners (3,1)(3, 1), (7,3)(7, 3), (5,7)(5, 7) and (1,5)(1, 5). Which of its sides are parallel? Explain how you know.

  1. Choose the pairs to test. Parallel sides never meet, so two sides that share a corner cannot be parallel. That leaves two pairs. The flatter sides run from (3, 1) to (7, 3) and from (1, 5) to (5, 7). The steeper sides run from (3, 1) to (1, 5) and from (7, 3) to (5, 7).

  2. Test the flatter sides. Measure straight up in two places. The lower one passes through (3, 1) and (5, 2). The upper one passes through (3, 6) and (5, 7). From (3, 1) up to (3, 6) is 5 units. From (5, 2) up to (5, 7) is 5 units. The gap stays the same, so the flatter sides are parallel.

    6−1=5and7−2=56 - 1 = 5 \quad \text{and} \quad 7 - 2 = 5

  3. Test the steeper sides. Measure straight across in two places. The left one passes through (2, 3) and (1, 5). The right one passes through (7, 3) and (6, 5). From (2, 3) across to (7, 3) is 5 units. From (1, 5) across to (6, 5) is 5 units. The gap stays the same, so the steeper sides are parallel too.

    7−2=5and6−1=57 - 2 = 5 \quad \text{and} \quad 6 - 1 = 5

Answer

The quadrilateral has two pairs of parallel sides: the two flatter sides and the two steeper sides. In each pair, the gap along the grid lines stays at 55 units, so the sides never get closer and would never meet.

The big idea. A polygon is a closed figure made of straight sides, and its number of sides gives its name. To say more about a polygon, compare its sides in pairs. Parallel sides stay the same distance apart and would never meet. Perpendicular sides meet at a right angle. When you explain why two sides are or are not parallel, talk about the gap between them, not about which way the figure is turned.

Next, you will classify triangles in two ways: by their angles and by the lengths of their sides.

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Classify Triangles by Angles and Sides

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