---
title: "Lines and Polygons"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 26
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons
---

# Chapter 26 — Lines and Polygons

With a ruler, a set square and a compass, geometry becomes a craft. This chapter teaches the two special positions of lines — [perpendicular](#def-g4-geometry-def) and [parallel](#def-g4-geometry-def) — at the tool level, and takes a tour of the [polygon](#def-g4-geometry-polygon) family. (The theory of these notions comes in [Chapter 40](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#ch-g6-lines).)

## 26.1 Perpendicular and parallel

**Definition 26.1 (Perpendicular, parallel).**

Two lines are *perpendicular* when they cross at a [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) (checked with the set square, marked with a little square). Two lines are *parallel* when they run in the same direction and never cross — like the two rails of a track, always at the same distance from each other.

**Method 26.2 (Drawing with the set square).**

*[Perpendicular](#def-g4-geometry-def) to a line $d$ through a point $P$*: lay one edge of the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) along $d$, slide until the other edge reaches $P$, draw.

*[Parallel](#def-g4-geometry-def) to $d$ through $P$*: lay the set square’s edge along $d$, press the ruler against another edge as a rail, slide the set square along the ruler up to $P$, draw.

![The two constructions. Parallel lines keep a constant gap: measuring the gap at two places is a good check.](https://one-course.com/images/onecourse/chapters/math-1/g4-geometry/fig-fad5bdff2884.svg)

*The two constructions. [Parallel lines](#def-g4-geometry-def) keep a constant gap: measuring the gap at two places is a good check.*

**Example 26.3 (Spotting them in figures).**

In a rectangle, each side is [perpendicular](#def-g4-geometry-def) to its two neighbors and [parallel](#def-g4-geometry-def) to the opposite side. Grid paper is a ready-made network of [parallels](#def-g4-geometry-def) and [perpendiculars](#def-g4-geometry-def) — that is what makes it so comfortable to draw on.

## 26.2 Polygons

**Definition 26.4 (Polygon).**

A *polygon* is a closed figure whose border is made of straight segments, its *sides*; the corner points are the *vertices*. Polygons are named by their number of sides: *triangle* ($3$), *quadrilateral* ($4$), *pentagon* ($5$), *hexagon* ($6$), *octagon* ($8$).

![The polygon family (the pentagon and hexagon drawn here are regular: equal sides and equal corners). A circle is not a polygon: its border is not made of segments.](https://one-course.com/images/onecourse/chapters/math-1/g4-geometry/fig-7aee915e5088.svg)

*The [polygon](#def-g4-geometry-polygon) family (the [pentagon](#def-g4-geometry-polygon) and [hexagon](#def-g4-geometry-polygon) drawn here are *regular*: equal sides and equal corners). A circle is not a [polygon](#def-g4-geometry-polygon): its border is not made of segments.*

**Example 26.5 (Describing a polygon).**

A useful game: describe a figure so precisely that a classmate can redraw it without seeing it. “A [quadrilateral](#def-g4-geometry-polygon) $ABCD$ with $AB = 5$ cm, the angle at $A$ right, $AD = 3$ cm …” — vocabulary is what makes the message work.

## 26.3 The circle and the compass

**Definition 26.6 (Circle).**

The *circle* of center $O$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm is the set of all points at exactly $3$ cm from $O$. The compass draws it: spike on $O$, opening $3$ cm, turn. A *diameter* is a segment through the center joining two points of the circle; it measures twice the [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes).

**Example 26.7 (The compass as a distance-keeper).**

The compass does more than draw circles: it *transports distances*. To mark all points at $4$ cm from a point $A$, draw the circle of center $A$, [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm; to compare two segments without a ruler, open the compass on one and test it on the other.

**Method 26.8 (Reproducing a figure on a grid).**

1. Choose a starting [vertex](#def-g4-geometry-polygon) and its position on the blank grid;
2. travel along the figure [vertex](#def-g4-geometry-polygon) by [vertex](#def-g4-geometry-polygon) , counting the grid steps (“ $3$ right, $2$ up”);
3. join the [vertices](#def-g4-geometry-polygon) in order, and compare with the model: same lengths, same directions.

## 26.4 Exercises

**Exercise 26.1 ★.**

Draw a line $d$ and a point $P$ not on it. Construct the [perpendicular](#def-g4-geometry-def) to $d$ through $P$, then the [parallel](#def-g4-geometry-def) to $d$ through $P$ (with the ruler-rail method). Mark the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle).

**Solution of Exercise 26.1.**

Constructions with [Method 26.2](#met-g4-geometry-draw); the little square marks the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) between $d$ and the [perpendicular](#def-g4-geometry-def).

**Exercise 26.2 ★.**

Look at the capital letters E, H, N, T, Z. In each, find a pair of [parallel](#def-g4-geometry-def) strokes and a pair of [perpendicular](#def-g4-geometry-def) strokes when they exist.

**Solution of Exercise 26.2.**

E: the three horizontal strokes are [parallel](#def-g4-geometry-def); each is [perpendicular](#def-g4-geometry-def) to the vertical one. H: the two vertical strokes are [parallel](#def-g4-geometry-def), the crossbar is [perpendicular](#def-g4-geometry-def) to both. N: the two vertical strokes are [parallel](#def-g4-geometry-def); the diagonal is neither. T: the two strokes are [perpendicular](#def-g4-geometry-def). Z: the top and bottom strokes are [parallel](#def-g4-geometry-def); the diagonal is neither.

**Exercise 26.3 ★.**

Name each [polygon](#def-g4-geometry-polygon): $6$ sides; $4$ sides; $3$ sides; $8$ sides; $5$ sides. How many *[vertices](#def-g4-geometry-polygon)* does each one have?

**Solution of Exercise 26.3.**

[Hexagon](#def-g4-geometry-polygon); [quadrilateral](#def-g4-geometry-polygon); triangle; [octagon](#def-g4-geometry-polygon); [pentagon](#def-g4-geometry-polygon). A [polygon](#def-g4-geometry-polygon) has as many [vertices](#def-g4-geometry-polygon) as sides: $6$, $4$, $3$, $8$, $5$.

**Exercise 26.4 ★.**

Draw any [pentagon](#def-g4-geometry-polygon) (not necessarily regular) and label its [vertices](#def-g4-geometry-polygon) $A$, $B$, $C$, $D$, $E$. How many sides does it have? Draw its diagonals from $A$ (segments joining $A$ to non-neighbor [vertices](#def-g4-geometry-polygon)): how many are there?

**Solution of Exercise 26.4.**

A [pentagon](#def-g4-geometry-polygon) has $5$ sides. From $A$, the non-neighbors are $C$ and $D$: two diagonals, $[AC]$ and $[AD]$.

**Exercise 26.5 ★.**

Draw a circle of center $O$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm; a [diameter](#def-g4-geometry-circle) $[AB]$; and a point $M$ of the circle not on $[AB]$. Give the lengths $OA$, $OM$ and $AB$ without measuring.

**Solution of Exercise 26.5.**

$OA = 4$ cm ([radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes)); $OM = 4$ cm (every point of the circle); $AB = 8$ cm (a [diameter](#def-g4-geometry-circle) is twice the [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes)).

**Exercise 26.6 ★.**

With the compass as a distance-keeper ([Example 26.7](#ex-g4-geometry-compass)), compare the sides of a triangle drawn by a classmate and decide whether it has two equal sides.

**Solution of Exercise 26.6.**

Open the compass along one side, then carry that opening to the two other sides: if the spike-and-pencil span fits a second side exactly, those two sides are equal.

**Exercise 26.7 ★.**

On grid paper, reproduce this path and close it into a [polygon](#def-g4-geometry-polygon): start at a point $A$; go $4$ right; $2$ up; $3$ left; $2$ up; then back to $A$ in a straight line. What is the name of the [polygon](#def-g4-geometry-polygon) you get (count its sides)?

**Solution of Exercise 26.7.**

The path closes into a [polygon](#def-g4-geometry-polygon) with $5$ sides — a [pentagon](#def-g4-geometry-polygon) (not a regular one!).

**Exercise 26.8 ★.**

True or false? Explain with a drawing. “Two [perpendicular lines](#def-g4-geometry-def) always cross.” “Two [parallel lines](#def-g4-geometry-def) never cross.” “A line can be [parallel](#def-g4-geometry-def) to one line and [perpendicular](#def-g4-geometry-def) to another.”

**Solution of Exercise 26.8.**

“[Perpendicular lines](#def-g4-geometry-def) always cross”: true — they cross at the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) itself.

“[Parallel lines](#def-g4-geometry-def) never cross”: true — that is their definition.

“A line can be [parallel](#def-g4-geometry-def) to one line and [perpendicular](#def-g4-geometry-def) to another”: true — on grid paper, a horizontal line is [parallel](#def-g4-geometry-def) to every horizontal line and [perpendicular](#def-g4-geometry-def) to every vertical one.

**Exercise 26.9 ★★.**

Draw a line $d$, then a line $e$ [perpendicular](#def-g4-geometry-def) to $d$, then a line $f$ [perpendicular](#def-g4-geometry-def) to $e$. Look at $d$ and $f$: what do you notice? (This observation becomes a theorem in [Chapter 40](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#ch-g6-lines).)

**Solution of Exercise 26.9.**

The lines $d$ and $f$ look [parallel](#def-g4-geometry-def) — and they always are: two lines [perpendicular](#def-g4-geometry-def) to the same line are [parallel](#def-g4-geometry-def) ([Proposition 40.3](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#prop-g6-lines-facts) makes this official).

**Exercise 26.10 ★★.**

Describe the figure below in words for a classmate, without showing it: a square of side $4$ cm, with a circle of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $2$ cm centered at the middle of the square (touching the four sides). Write the description as numbered instructions.

**Solution of Exercise 26.10.**

One correct program: “1. Draw a square $ABCD$ of side $4$ cm. 2. Mark the point $O$ where its diagonals cross. 3. Draw the circle of center $O$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $2$ cm.” The circle touches each side at its middle.

**Exercise 26.11 ★★.**

How many diagonals does a [quadrilateral](#def-g4-geometry-polygon) have? A [pentagon](#def-g4-geometry-polygon)? A [hexagon](#def-g4-geometry-polygon)? Draw the three cases, count carefully, and describe the pattern you see.

**Solution of Exercise 26.11.**

[Quadrilateral](#def-g4-geometry-polygon): $2$ diagonals. [Pentagon](#def-g4-geometry-polygon): $5$. [Hexagon](#def-g4-geometry-polygon): $9$. Each new [vertex](#def-g4-geometry-polygon) adds more and more diagonals — the counts grow faster and faster ($+3$, then $+4$, …).