Lesson video

Hello, welcome one and all to this lesson on constructions with me, Mr. Gratton.

Grab a pair of compasses and a ruler in this lesson where we will construct a variety of different angles, including 90-degree and 45-degree angles and beyond.

Pause here to have a look at some important keywords that might come in helpful during this lesson.

First up, let's take some important construction techniques and apply them to constructing some 90-degree and 45-degree angles.

Let's have a look.

Here we have Lucas who is familiar with constructing a perpendicular bisector, a perpendicular to a line segment through a specific point and an angled bisector.

Lucas wonders whether we can combine multiple constructions to create some different shapes and angles, and if we can combine them, how would we do so? Let's see if we can combine different constructions by first looking at constructing a perpendicular through a point.

We can do this by drawing a large circle through the point we want to pass the perpendicular through or simply the arcs of a circle instead, and then create a modified line segment by shortening and extending the line segment so that the new line segment lies fully within the circle or the two circular arcs that we just drew.

We have shortened this line segment on its left and extended it on its right.

We can then construct the perpendicular bisector to our modified line segment that passes through our point.

That perpendicular line creates a right angle with our original line segment.

For this same construction, we can then dissect the 90-degree angle in order to make a 45-degree angle.

When applying multiple sets of constructions, it can be helpful to extend line segments to ensure that any arcs and circles intersect correctly.

With the compass needle on the vertex of that right angle, we can create two equally long legs where the two intersections are the endpoints of those legs, like so.

From these two endpoints, construct two more arcs here and here with the same radius as the original two arcs in order to create one more point of intersection.

Joining that point to the vertex of the 90-degree angle bisects the 90-degree angle to create a 45-degree angle.

We can then use this full construction to also find a different 90-degree angle, a 135-degree angle as the supplementary angle to the 45-degree one that we've highlighted, and also a 225-degree angle here.

Pause here to think about or discuss what other angles can you find on this diagram? Okay, here's a check.

Here we have a line segment AB and a point K.

Pause here to consider which of these three constructions will result in a 90-degree angle.

Remember, a line that creates a 90-degree angle is also called a perpendicular.

So either a perpendicular that bisects line segment AB, or a perpendicular through K will result in a 90-degree angle.

So both A and B are correct, like so.

We've now produced a perpendicular through point K.

The point of intersection between the perpendicular and line segment AB is point P.

Pause again to consider which of these constructions produces an angle of 45 degrees.

In order to get a 45-degree angle, we have to bisect a 90-degree angle.

Both APK and KPB are 90-degree angles, which when halved therefore give 45-degree angles.

So we've decided to bisect APK.

With this set of constructions, pause here to match the angles with their sizes.

Great stuff.

Here are the answers.

Pause here to compare your answers to these on screen.

Okay, let's go through a construction together.

This construction aims to make a right angle triangle with a hypotenuse of a certain fixed length.

In my example on the left, I have drawn a six-centimeter line segment.

This will be the hypotenuse of my right angle triangle.

One of the shorter sides will be a part of this ray that can be of any length.

Pause now to draw a four-centimeter line segment and a ray of any length longer than four centimetres on one of the two endpoints of your four-centimeter line segment.

Okay, our goal, first of all, is to construct a right angle.

To do this, we can construct a perpendicular to AB.

If our perpendicular also passes through that open endpoint of the hypotenuse that we drew earlier, we will bound a right angled triangle.

So let's go through the steps to draw a perpendicular through that point, through the open endpoint of the soon-to-be hypothesis.

First up, place the compass needle on that open endpoint and set the compass width so that the circle or arcs intersect with AB twice.

Pause now to try this for yourself and remember, there is no one correct compass width as long as the two arcs or the circle intersect AB twice.

So we have two intersections on the line segment AB.

At each intersection, construct two more arcs, each with an equal radius to each other, like so, for a total of four arcs that intersect twice.

Pause now to try this for yourself.

This is what you should have.

Remember, depending on the radius of the arcs that you've drawn, you might have slightly different locations for your two intersections.

We can then draw a line segment through those two intersections.

This perpendicular passes through the open endpoint of the hypotenuse.

Pause now to try this for yourself.

This is the result.

There are a lot of constructions here.

Amongst them, we have created a right angle triangle.

In my case, this right angle triangle has a hypotenuse of six centimetres.

Pause here to identify the right angle triangle in your construction and then use a protractor to check it really is as close to a right angle as possible.

Lovely stuff.

Onto some independent practise.

For question one, let's draw two different perpendiculars to AB.

One is the perpendicular bisector and the other is a perpendicular through point K.

And for question two, construct an angle bisector to DHE and then consider the size of some of the angles that you've constructed.

Pause now for these two questions.

For questions three and four, here we have two identical diagrams. A line segment of seven units and a ray of some longer length are drawn.

Pause here to construct two different right angle triangles, one with a hypotenuse of seven units and a different one with a non-hypotenuse of seven units instead.

And for question five, from point A, a boat starts travelling at a bearing of 72 degrees for six kilometres.

The boat then turns to a different bearing of 162 degrees and travels a further 2.

5 kilometres.

Construct the journey of this boat starting from point A and then find the distance AC and angle BAC.

Pause now to do question five.

Great work on all of those constructions.

Onto the answers.

Here are questions one and two.

For question two, angle DHJ is 45 degrees, angle JHF is 135 degrees and the reflex angle GHJ is 225 degrees.

Pause here to compare all of your constructions for questions one and two to those on screen.

For questions three and four, pause here to check your construction lines and shapes of these two right angle triangles.

And finally, question five, six kilometres from point A is point B.

The triangle made by points A, B, C is a right-angled triangle.

Pause here to check this complete construction where the distance from A to C on the construction is 6.

5 centimetres, meaning its real distance is 6.

5 kilometres.

Furthermore, angle BAC is 22.

6 degrees.

Great work on all of those constructions, but let's not restrict ourselves to just 45 and 90-degree angles.

Let's use the constructions we've done so far and the knowledge of 45 and 90-degree angles to help us construct a range of different angles as well.

Here we have Lucas who thinks there's not much that we can do with a 40-degree angle.

However, Laura disagrees.

Laura thinks that we can make a range of different angles related to 40 degrees.

The most simple of these is a 140-degree supplementary angle by extending one of the legs of the 40 degree angle like so.

Pause here to think about or discuss if there are any other angles that we can make from just a 40-degree angle using some different constructions.

Here are some examples of what we can do with a 40-degree angle.

We have an angle bisector that can create a 20-degree angle.

A perpendicular through the vertex of the angle can create a 50-degree angle.

Notice how the 40-degree angle must be complemented by a 50-degree angle to make a right angle of 90 degrees.

We can also perform a two-stage construction, an angle bisector and then a perpendicular through the angle vertex to create a 70-degree angle here.

We can also perform an angle bisector through the reflex angle of 320 degrees to create a 160-degree angle as well.

Okay, here's a series of checks.

For this first check, these two line segments are the legs of a 100-degree angle.

Pause here to consider, which of these constructions creates a 50-degree angle? An angle bisector halves the angle from 100 degrees to 50 degrees.

For this same 100-degree angle, pause here to consider which of these constructions creates an 80-degree angle.

We can extend the line segment AB like so to create a supplementary angle of 80 degrees.

This is because the angle ABC is 100 degrees and so 80 degrees is what's left over, the supplementary angle to a straight line of 180 degrees.

Next up, pause here to identify the construction that creates a 10-degree angle.

A perpendicular to AB through the point B splits this 100-degree angle up into a 90-degree right angle and then that little remaining angle is 10 degrees.

Okay, sticking with the same angle, but let's get a little bit more tricky.

Pause here to identify the two constructions that, when put together, create a 55-degree angle.

Write down your two answers in the correct order.

We have B, a perpendicular, and then C, an angle bisector.

The perpendicular gives us a 90-degree angle and a 10-degree angle.

We can then construct an angle bisector between the line segments AB and that perpendicular.

This then bisects the 90-degree angle to create a 45-degree angle.

When we then add on that extra 10-degree angle adjacent to the 45-degree angle, we create a 45 plus 10-degree angle, which is 55 degrees.

A very well done if you spotted this compound construction.

And lastly, line segments DE and EF are the legs of a 130-degree angle.

Let's add a bunch of different constructions, an angle bisector and a perpendicular through point E.

Pause here to consider the impact of these two constructions on the angle DEF, that 130-degree angle we originally had.

And hence find the size of the angles W degrees, X degrees, Y degrees and Z degrees.

W degrees is 65 degrees because it is a bisection of the 130-degree angle and half of 130 is 65.

X degrees is 25 degrees and y degrees is 40 degrees because they sum to the same 65 degrees, which we got as a bisection of the 130-degree angle.

25 degrees comes from the remainder of 65 degrees and the 90 degrees that we got from the perpendicular.

And lastly, because x plus y plus z equals 180, z degrees equals 180 degrees, take away 25 degrees, take away 40 degrees, which is 115 degrees.

Brilliant.

Onto the practise questions for task B.

For each of these practise questions, check the accuracy of the angles in your constructions using a protractor.

That way you can check whether your constructions are correct straightaway.

However, do not use the protractor to aid in any of the actual constructions of any of your angles.

Brilliant stuff, onto questions one and two.

Pause here to create a 71 degree, 109 degree, 38 degree, and 19 degree angle from these two diagrams. And pause here once more to create the angles 30 degrees, 15 degrees, eight degrees, and 24 degrees.

And lastly, each of these are 32-degree angles.

Create as many of these angles from just this 32-degree angle as you can.

If you need more copies of a 32-degree angle, use your protractor to create another 32 degrees.

Pause now to do this final question.

Great effort on exploring all of those angles.

For question one, we can construct a perpendicular at the vertex of the angle to create a 71-degree angle and then we can extend one of the original legs of the angle to create a 109-degree angle with that perpendicular.

For question two, 38 degrees comes from the bisection of the 76-degree angle and 19 degrees comes from a bisection of one of the two 38-degree angles.

Pause here to see whether these constructions match your own.

For question three, the 60-degree angle can be used as an angle adjacent to the hypotenuse of a right angle triangle.

An angle opposite the 60-degree angle is a 30-degree angle that can be bisected to make a 15-degree angle.

For question four, a perpendicular at the vertex of the 82-degree angle will give an 8-degree angle.

Furthermore, bisecting the third interior angle of the triangle gives a 24-degree angle.

Pause here to see whether these constructions match your own.

And for question five, we have a 16-degree angle from a bisector, a 58-degree angle from a perpendicular, a 29-degree angle from a bisection.

And here we have a 24-degree angle from a bisection and some angles added together, and we've also got a 148-degree angle by extending that line segment.

We also have a 74-degree angle by bisecting that 148-degree angle.

Furthermore, we have a 164-degree angle by extending a line segment and then dissecting that 164-degree angle gives you an 82-degree angle.

We also have an eight-degree angle by creating a perpendicular, and here we have a 328-degree angle.

We also have a 238-degree angle by creating a perpendicular, and I recommend that you pause here to check some of your angles against these on screen.

A very well done for all of the angles you managed to create from just one single 32-degree angle.

Great job, everyone in bringing together this massive range of constructions and angles in a lesson where we have constructed perpendiculars to create 90-degree angles and then bisected those angles to get to 45-degree angles.

Furthermore, we can use our wide range of construction techniques to make a variety of different angles if given any angle x degrees.

For example, we can extend line segments, we can construct perpendiculars through vertices of an angle or we could bisect that angle.

Furthermore, we can use any combination of these techniques to create even more different angles.

Once again, thank you all so much for your great work and effort during today's lesson.

I've been Mr. Gratton and you have all been simply fantastic.

until our next maths lesson together, take care, everyone and goodbye.