Right Angles, Parallel and Perpendicular Lines

You can draw the edge of a tile or show a path that keeps going. The ends of your drawing tell these ideas apart.

Mark a place

A point is an exact place. Draw a small dot and label it with a capital letter. You can call it "point M." The dot helps you see the place. The point itself has no length or width.

Look at the ends

A line is straight and goes on forever both ways. Your page shows just part of it. Draw an arrowhead at each end to show that it keeps going. Through two different points, you can draw exactly one straight line.

A line segment starts and stops at two endpoints. It has a length you can measure. Each straight side of a shape is a segment.

A ray starts at one endpoint and keeps going in one direction. Draw a dot at its start and an arrowhead at its other end. A flashlight beam helps you picture that one-way path.

Draw and name each kind

Mark points P and Q. Use a ruler to join them. Stop at both points. You have segment PQ. Calling it "segment QP" names the same segment.

To show line PQ, extend the straight drawing past both points. Add arrowheads at both ends. "Line QP" names the same line.

For ray PQ, begin at P, go through Q, and continue past Q. Add one arrowhead beyond Q. The first letter names the endpoint. Ray QP starts at Q and goes through P, so it is a different ray.

Some books use these symbols:

• Line AB: $\overleftrightarrow{AB}$
• Segment AB: $\overline{AB}$
• Ray AB: $\overrightarrow{AB}$

The marks above the letters match the ends of each drawing.

A drawing is only a piece

A line drawn across a small card keeps going just as far as a line drawn across a poster: forever. You cannot decide that one line is longer from its drawing. You can compare the lengths of segments, because they stop.

Use your ruler for straight paths. A path that bends is a curve. In math, a line is always straight.

Look at a page held the usual way for reading. Its edges help you describe how a straight line runs.

Across and up

A horizontal line runs straight across, level with the ground. Think of still water in a glass. On your page, it runs the same way as the top and bottom edges. "Horizontal" comes from "horizon," where the sky seems to meet the sea.

A vertical line runs straight up and down. Think of a string hanging with a weight at its end. On your page, a vertical line runs the same way as the left and right edges.

A slanted line is neither horizontal nor vertical. It tilts, like the straight side of a roof. Another word for slanted is oblique. People also use the word "diagonal."

Read the strokes of a letter

Use plain capital letters made with straight strokes.

• In "T," the top stroke is horizontal. The stem is vertical.
• In "N," both side strokes are vertical. The middle stroke is slanted.
• In "Z," the top and bottom strokes are horizontal. Its middle stroke is slanted.
• In "A," the long strokes slant. The short crossbar is horizontal.

Now trace the middle stroke of "N" with your finger. It travels across and down at the same time. That helps you see its slant.

Draw on grid paper

Grid paper has rows of little squares. Place your ruler along a grid line that runs across. Draw along it to make a horizontal segment. Use an up-and-down grid line to make a vertical segment. Join two grid points across and up to make a slanted segment.

Length does not decide the name. A short vertical segment still runs up and down. A very long segment that tilts a little is slanted, even if it looks nearly flat.

Turn the page

Draw a horizontal segment. Keep your eyes facing the same way and turn the page a quarter turn. The segment now runs up and down: it is vertical. A vertical segment becomes horizontal after that turn. A slanted segment stays slanted after a quarter turn.

Two different horizontal lines never meet. Two different vertical lines never meet either. A horizontal line and a vertical line make a square corner where they cross.

Open a book a little, then open it wider. The turning point stays in the same place. The amount of turn changes.

Find the corner and arms

An angle is made where two straight arms meet. The arms can be rays or segments. Their meeting point is the vertex. More than one vertex are called vertices.

Think of a door turning on its hinges. The hinge side gives you the turning place. Picture one arm staying still while the other turns around that place. A bigger turn makes a bigger angle.

You can also find angles between clock hands, at your bent elbow, and inside the corners of flat shapes. A triangle has 3 angles, a square has 4, and a pentagon has 5. A circle has no corners or straight sides, so it has no angles.

Size comes from the turn

Draw an angle with short arms. Trace it on see-through paper. Make the traced arms three times as long without changing their directions.

Place the traced vertex on the first vertex. Line up one arm. The other arms line up too. Both angles have the same opening, even though one drawing stretches farther across the page.

The gap between the tips can grow when arms get longer. That gap does not measure the turn. Look near the vertex to compare the opening.

Show the opening you mean

Two arms have an opening on each side. A small curved mark shows which opening you mean. Without a mark, use the smaller opening. When the arms point in opposite directions, both openings are the same size: a half turn.

To name an angle, use three capital letters. Put the vertex letter in the middle. If the arms meet at B and pass through A and C, write ∠ABC or ∠CBA. Say "angle ABC" or "angle CBA." Both names keep B in the middle.

Choose the way to turn

Clockwise means the way clock hands move. Counterclockwise means the other way.

Face the front of a room. A quarter turn clockwise leaves you facing one side wall. A quarter turn counterclockwise leaves you facing the other side wall. Both turns have the same size. They go in different directions.

A half turn either way leaves you facing the back. The amount of turn and the direction of turn are two separate things to describe.

Stand facing a wall. Turn to face the next wall beside it. You have made a quarter turn. Four equal turns like that bring you all the way around.

Count with square corners

A quarter turn makes one right angle. A right angle is a square corner. The word "right" names the corner, so a quarter turn to the left makes one too.

A whole turn is also called a full turn. A half turn does not bring you back to your starting way. It brings you halfway around.

After a half turn, the starting arm and ending arm point opposite ways. Together they make a straight line. This opening is a straight angle, equal to two right angles.

Follow a clock hand

Watch the minute hand start at 12 and travel clockwise.

1. At 3, it has made a quarter turn in 15 minutes.
2. At 6, it has made a half turn in 30 minutes.
3. At 9, it has made a three-quarter turn in 45 minutes.
4. Back at 12, it has made a whole turn in 60 minutes.

Each jump of three clock numbers is one right angle. At exactly 3 o'clock or 9 o'clock, the two hands make a right angle. At 6 o'clock, they make a straight angle.

Turn through compass directions

The four main directions go north, east, south, west, then north again clockwise. Face north. One quarter turn clockwise takes you east. A second takes you south. A third takes you west. A fourth takes you north again.

You can reach west with a shorter turn too: one quarter turn counterclockwise from north. Ending in the same direction does not mean making the same amount of turn.

Join turns going the same way

A toy robot faces a door. It turns a quarter turn clockwise, then a half turn clockwise. Count one right angle, then two more:

It has made a three-quarter turn clockwise.

The corner of a book is a useful model of a right angle. You can carry a paper version around to check other corners.

Make your checker

You can use a corner of a rectangular card. Or make a right-angle checker from scrap paper, even with torn edges.

1. Fold the paper once to make a straight folded edge.
2. Fold it again so that this folded edge lies exactly on itself.
3. Find the new corner where the two folds meet.

That corner is a right angle.

Match the opening

Put the checker's corner on the angle's vertex, inside the opening. Line up one checker edge with one arm. Look at the other arm. If it lies exactly along the other checker edge, the angle is right.

A small square mark in a drawing tells you that an angle is right. A curved mark simply shows which opening to look at.

Turn your checker to match a tilted angle. A right angle stays right when you turn, tip, or flip it. One arm does not need to run across the page.

Count corners inside a shape

Start at one corner and move around the shape in order. Check each opening from inside the shape. Mark the right angles as you go, so you do not count one twice.

A square and a rectangle each have 4 right angles. A triangle can have at most 1. A triangle with one right angle is called a right triangle. A regular hexagon has no right angles.

An L-shaped outline made by removing a corner rectangle from a rectangle has 6 corners. Five inside openings are right angles. At the dent, the inside opening wraps around more than a half turn. The little square gap outside the shape is not its inside angle. This L shape has 5 right angles.

Look at letter strokes

Picture thin, straight strokes in plain block capitals. Count the square openings where strokes meet, not corners around thick letter outlines.

• "L" has 1 right angle.
• "T" has 2, one on each side of its stem.
• "F" has 3.
• "E" and "H" each have 4.

A corner can look nearly square and still be a little too wide or narrow. Slide your checker into place to find out. Two straight lines that meet at a right angle are called perpendicular.

An angle can open less than a paper corner or more than one. A right angle helps you name and compare these openings.

Start with the marked opening

Find the curved mark. If there is no mark, use the smaller opening. Put your checker inside that opening, with its corner on the vertex and one edge along an arm.

An arm inside the checker's corner makes an opening smaller than a right angle. An exact match makes a right angle. An opening reaching past the other edge is bigger. An opening of more than a half turn is bigger than a right angle too.

Give the opening a name

A narrow pizza tip can show an acute angle. A door opened past a square corner, but short of a straight line, shows an obtuse angle. A straight angle is exactly two right angles. It is not obtuse, because obtuse stops short of a half turn.

The minute hand traveling clockwise from 12 to 9 turns through a reflex angle. That turn is three right angles. The smaller opening between the start and finish is different, so keep track of the opening you mean.

Put angles in order

On a clock, look at the opening between the hands that is no more than a half turn. At 2 o'clock it is acute. At 3 it is right. At 5 it is obtuse. At 6 it is straight. Each jump of one clock number is one third of a right angle.

From smallest to biggest, the angle types go acute, right, obtuse, straight, reflex, then a whole turn.

To compare two acute angles, trace one onto see-through paper. Place its vertex on the other vertex and line up one arm. The second arm that has turned farther shows the bigger opening. Arm lengths do not decide the result.

Find angle types in shapes

A triangle with three equal sides has three acute angles. Every triangle has at least two acute angles. A regular hexagon has six obtuse angles. A four-sided shape with two pairs of parallel sides, but no square corners, has two acute and two obtuse angles.

An angle just short of a right angle is acute.

You can make an angle by turning two paper strips apart. Keep them joined at one end. That end is the vertex.

Feel the opening grow

Start with the strips closed together. Turn one strip while holding the other still. Before a quarter turn, the opening is acute. At a quarter turn, it is right. Past a quarter turn but before a half turn, it is obtuse. Keep going and you reach straight, then reflex angles.

Draw a right angle

Use a sharp pencil, a ruler, and your right-angle checker. On grid paper:

1. Choose a point where grid lines cross. Label it V.
2. Draw one arm from V along a horizontal grid line.
3. Draw the other arm from V along a vertical grid line.
4. Add a small square mark inside the corner.

Grid lines meet at right angles. Both arms must begin at the same point.

On plain paper, draw the first arm with a ruler. Put a card's corner on its endpoint. Line up one card edge with the arm. Draw the second arm along the card's other edge.

Draw a smaller or wider opening

Draw a first arm from V. Picture turning away from it around V. Stop before a quarter turn for an acute angle. Stop after a quarter turn but before a half turn for an obtuse angle. Use a ruler to draw the second arm. Add a curved mark to show your opening.

Try this on grid paper. Draw the first arm 4 squares right from V. Then choose one second arm:

Draw each one separately so you can see its opening clearly. The acute arm stays inside the paper corner. The obtuse opening goes past that corner, but stops short of a straight line.

Check your pencil work

Place the checker on V and along the first arm. Test the opening, then label it "acute," "right," or "obtuse." Use the ruler again if an arm wobbles. If there is a gap at V, redraw the arm so the two arms share their corner.

For another drawing, point the first arm in a new direction. Turn the checker to fit it before drawing the next arm.

Look at the two long edges of a ruler. They run beside each other without getting closer together. They show a pair of parallel segments.

Imagine the lines continuing

Parallel lines are straight lines on a flat surface that never meet. They stay the same distance apart all the way along. The lines on notebook paper show this, as do straight railroad tracks.

Two segments are parallel when the lines they lie on are parallel. Imagine extending both segments in both directions. Their lengths can be different. One can begin farther along than the other.

Not touching in a small drawing is not enough. Two lines may slowly get closer and meet beyond the edge of the page.

Check the gap

On grid paper, begin with two horizontal lines. Count straight across the gap, up or down, at one end. Count again farther along. A gap of three squares at both places helps you check that they stay apart evenly.

For slanted lines, compare gaps measured in the same direction. Also follow the lines across the drawing to check that they keep the same slant and never cross. Use your ruler to extend a short drawing when needed.

Parallel lines can be horizontal, vertical, or slanted. Any two different horizontal lines are parallel. Any two different vertical lines are parallel too.

Read the marks

The symbol ∥ means "is parallel to." Read "AB ∥ CD" as "line AB is parallel to line CD."

Matching small arrow marks on two lines also show parallel lines. These marks sit on the lines. Arrowheads at the ends of a line drawing mean that the line continues.

Find pairs in letters

Use plain block capitals with straight strokes. In "H," the two upright strokes are parallel. In "E," all three horizontal strokes are parallel to one another. In "Z," the top and bottom strokes are parallel.

The strokes of "V" meet at the bottom, so they are not parallel. The edges of a curved road are curves. Even when their gap stays even, they are not parallel straight lines.

Cross two pencils on a table. You can make a narrow opening or a square one. Both crossings show lines meeting, but only the square crossing is perpendicular.

Crossing lines

Intersecting lines meet at a point. That place is the point of intersection. Two different straight lines on a flat surface either never meet or meet at exactly one point.

Perpendicular lines meet at a right angle. Use your checker at the meeting point. Line up one checker edge with one line. If the other line follows the other edge, the lines are perpendicular.

Crossing alone is not enough. A narrow crossing gives acute and obtuse angles. It is intersecting, but not perpendicular.

Look around the crossing

When two lines cross at a right angle, all four openings are right angles. Together they make a whole turn.

When crossing lines are not perpendicular, there are two acute openings and two obtuse openings. The two acute ones face each other across the crossing. The two obtuse ones do too.

Ends can meet too

Segments can make square corners in several ways:

• A plus sign has two segments crossing through each other.
• A block "T" has one segment ending on another.
• A block "L" has two segments meeting at their ends.

These segments are perpendicular when they make right angles. Imagine continuing them into full lines: those lines cross at square corners.

Read and turn the drawing

The symbol ⊥ means "is perpendicular to." Read "AB ⊥ CD" as "line AB is perpendicular to line CD." A small square at their meeting point shows a right angle.

A horizontal line and a vertical line are perpendicular. Slanted lines can be perpendicular too. Turn a plus sign on your page until both strokes slant. The square corners stay the same.

Turning the page changes which way the lines face. It does not change whether the pair is parallel or perpendicular.

Grid paper helps you draw lines that stay apart and lines that make square corners. You can use its rows or count steps between grid points.

Start with grid lines

Draw a horizontal segment with a ruler. Draw another along a different horizontal grid line. The segments are parallel. For a perpendicular segment, draw along a vertical grid line.

You can also draw a parallel pair along two different vertical grid lines.

Copy a slant

Draw from one grid point to another 3 squares right and 1 up. Now choose a starting point off that line. From there, draw 3 right and 1 up again. These lines have the same slant, so they are parallel.

You could draw 6 right and 2 up for the second line. Those steps repeat 3 right and 1 up twice. Starting on the first line would draw over it, so choose a point away from it.

Make a square corner on a slant

For a line drawn 3 right and 1 up, swap the step numbers and change one direction. Draw 1 left and 3 up. These directions make perpendicular lines. Going 1 right and 3 down draws the same slant the other way.

Swapping numbers alone does not work. The steps 1 right and 3 up do not make a right angle with 3 right and 1 up. Check with your paper corner.

Build a tilted square

Use one grid point as A.

1. From A, go 2 right and 1 up to B. Draw AB.
2. From B, go 1 left and 2 up to C. Draw BC.
3. From A, go 1 left and 2 up to D. Draw AD.
4. Join D to C.

AB and BC are perpendicular. AD and BC are parallel. The final side DC has the same steps as AB. The four sides make a tilted square. Fit your checker into each corner to test it.

Draw on plain paper

Draw along both long edges of a ruler to make parallel lines. Keep the ruler still between the two strokes.

For perpendicular lines, place a card corner on the first line. Line up one card edge with that line. Draw along the other edge. You can turn the card to work with a slanted first line too.

A shape's sides are segments. To compare two sides, imagine the straight lines they lie on continuing beyond the corners.

Follow a rectangle

A rectangle has four right angles. Its opposite sides are parallel: top with bottom, and left with right. That gives 2 pairs of parallel sides.

Sides meeting at each corner are perpendicular. There are 4 pairs of perpendicular sides. Count pairs, not individual sides.

A square is a special rectangle with all four sides equal. It has the same line pairs and right angles. Turning it onto one corner does not change any of these facts.

Compare other shapes

Every pair of sides in a triangle meets at a corner, so triangles have no parallel sides. A right triangle's square corner gives one perpendicular pair.

A slanted four-sided shape with two pairs of parallel sides, but which is not a rectangle, has no right angles and no perpendicular sides.

Check a shape you draw

On grid paper, draw a bottom side 4 squares right. From its left endpoint, draw a left side 2 squares up. Draw the top side 2 squares right from there. Join the top's right endpoint to the bottom's right endpoint.

The top and bottom are horizontal, so they form one parallel pair. The vertical left side makes a right angle with each of them. There are two right angles. The slanted side is neither parallel nor perpendicular to another side.

Sides can be apart

Picture a regular eight-sided stop-sign shape with a flat top. Its top side and its rightmost vertical side are perpendicular, even though they do not touch. Extending them into lines shows a right-angle crossing.

This regular octagon has four parallel side pairs. None of its inside corners is a right angle. Perpendicular side directions and right angles inside a shape are different things to check.

Look around you

Straight ladder rungs show parallel segments. So do level shelves in a bookcase. The upright sides of a rectangular door frame are parallel to each other. Each is perpendicular to the top.

When sorting shapes, ask two separate questions: "Are there parallel sides?" and "Are there perpendicular sides?"