Turns, parallel lines and square corners

An angle measures an opening

Put two strips together at one end and open them like scissors. Their meeting point is the vertex. The angle is the opening between the two directions, not the length of either strip.

A square corner gives a right angle. To make a paper tester, fold a sheet once. Fold it again so that the two parts of the first folded edge lie exactly on top of each other. The corner where the folds meet is a right angle.

Put the tester’s corner at the vertex and line up one edge. Compare the other edge with the second arm. An opening narrower than the tester is smaller than a right angle; a wider opening is greater than a right angle. Extending an arm does not change the opening.

Compare the opening, not the arm lengths

The curved mark below shows which opening to compare. Place the corner of your paper tester at the meeting point and line up one edge with the horizontal arm.

Opening A

Two straight arms meet at the vertex. Compare the opening marked by the arc with your square corner.

View figure data

Positions for recreating the drawing; equal scales, y increasing upwards

Point	x	y
Vertex	0	0
First arm endpoint	6	0
Second arm endpoint	3	4

Positions for recreating the drawing; equal scales, y increasing upwards

Point	x	y
Vertex	0	0
First arm endpoint	6	0
Second arm endpoint	3	4

The second arm lies inside the tester’s right angle. Opening A is smaller than a right angle. Making the horizontal arm longer would not turn either direction, so it would not change the angle.

Check your understanding 1 Turn your tester with the angle

Compare the marked opening B with a right angle. Write smaller, right or greater before revealing the answer. Explain how your tester fits.

Opening B

The arms slope in different directions. Turn your square corner to line up one edge with an arm.

View figure data

Positions for recreating the drawing; equal scales, y increasing upwards

Point	x	y
Vertex	0	0
First arm endpoint	4	3
Second arm endpoint	-3	4

Positions for recreating the drawing; equal scales, y increasing upwards

Point	x	y
Vertex	0	0
First arm endpoint	4	3
Second arm endpoint	-3	4

Your comparison and explanation

Show answer

Right. Both arms line up with the edges of the tester. A square corner still fits when the whole drawing is turned; a right angle need not have a horizontal arm.

Check your understanding 2 Ignore the unequal arm lengths

Compare the marked opening C with a right angle. Line up your tester with either arm; keep its corner at the vertex.

Opening C

Two arms of different lengths meet. The arc marks the opening to compare.

View figure data

Positions for recreating the drawing; equal scales, y increasing upwards

Point	x	y
Vertex	2	3
First arm endpoint	0	0
Second arm endpoint	7	2

Positions for recreating the drawing; equal scales, y increasing upwards

Point	x	y
Vertex	2	3
First arm endpoint	0	0
Second arm endpoint	7	2

Your comparison and explanation

Show answer

Greater. With one edge lined up, the other arm is outside the tester’s right angle. The longer arm does not make the opening greater: the two directions do. If you extend the shorter arm along the same line, the comparison stays the same.

Make a smaller opening with two strips. Turn them together, then extend one strip along its direction. Explain why neither change turns your smaller angle into a right angle.

Perpendicular lines meet at a right angle

Two lines are perpendicular when they meet at a right angle. A vertical line and a horizontal line give one example, but turning the whole drawing keeps the right angle.

To draw a perpendicular through a point on a line, put one straight edge of a set square along the line with its square corner at the point. Draw along the other right-angle edge. Check the corner with your tester.

Parallel lines keep the same direction

Straight lines in a plane are parallel when they never meet, however far they are extended. The gap between them, measured square across, stays the same.

On squared paper, draw one line along a horizontal grid line and another two squares below it. They are parallel. Two short segments that do not touch are not automatically parallel: their extended lines might meet.

To draw another direction of parallel lines, keep a set square against a ruler and slide it without turning. Draw along the same edge before and after sliding.

Check your understanding 3 Decide from the whole lines

Two drawn line segments do not touch, but one slopes up to the right and the other slopes down to the right. Is “they do not touch” enough to call them parallel? Explain how to check.

Your answer or drawing on paper

Show answer

No. Extend their directions, or compare their perpendicular separation at different places. If the directions lead them to meet, the lines are not parallel even though the short drawn segments do not touch.

Draw, then inspect

Draw one pair of parallel lines and one pair of perpendicular lines in different orientations. Use your right-angle tester where the perpendicular lines meet. A person or a suitable construction check must inspect the drawing; writing the names does not verify it.

The given line below is sloping. Turn your tester or set square with the drawing; a right angle does not have to look like a horizontal line meeting a vertical line.

Try the model. You can change your answer or look at the worked model. This activity has no score.

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
g	(20, 40) → (80, 70)	Given
P	(40, 50), included	Given

The check looks for one straight line through P, perpendicular to g. It checks the drawing, not how you used a set square. You can also draw on paper and compare with the public answer.

Show solution

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
g	(20, 40) → (80, 70)	Given
P	(40, 50), included	Given
example	(30, 70) → (50, 30); start unspecified, end unspecified	Drawn

For the next drawing, the point is away from the given line. Draw through P in the same direction as g. A line drawn on top of g would share all its points and would not be a distinct parallel line.

Try the model. You can change your answer or look at the worked model. This activity has no score.

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
g	(20, 40) → (80, 70)	Given
P	(40, 80), included	Given

The check looks for one straight line through P, parallel to and separate from g. It checks the finished drawing. You can also draw on paper and compare with the public answer.

Show solution

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
g	(20, 40) → (80, 70)	Given
P	(40, 80), included	Given
example	(20, 70) → (60, 90); start unspecified, end unspecified	Drawn

Shorten your correct line or reverse its two endpoints. The relationship should stay the same as long as your drawn segment still goes through P. Then draw on paper in another orientation and check with your tools.