Lesson video

Hello there.

You made a great choice with today's lesson.

It's gonna be a good one.

My name is Dr.

Robson and I'm gonna be supporting you through it.

Let's get started.

Welcome to today's lesson from the unit of angles.

This lesson is called Problem Solving with angles, and by the end of today's lesson we'll be able to use our knowledge of angles to solve problems. Here are some previous keywords that may be useful during today's lesson, so you may want to pause the video if you want to remind yourself what any of these words mean and then press play when you're ready to continue.

This lesson is broken into two learning cycles where we'll be looking at problems involving angles in each learning cycle.

In the second learning cycle, these problems will be more complex than in the first one, but let's start off with our first set of problems with angles.

When solving a problem involving angles, you might not be able to find the angle you want straight away.

Sometimes you may need to find other unknown angles before you can find the one that you are asked for.

But the good news is that the more angles you find, the easier it can be to find other angles.

You may need to bear in mind that you may need to use multiple angle facts in order to solve a problem and the angle facts that you use may vary between problems. It all depends on what information you are presented with in the first place.

Sometimes when you see a problem, you might see something that looks like a line segment.

In other words, something that looks straight, but, just because something looks like a line segment doesn't necessarily mean it is one.

It might not be exactly straight, it might be two line segments joined together.

An angle that is very close to 180 degrees, but not quite.

Usually a problem will clarify whether or not this is a case somewhere within its information.

Throughout this lesson you may assume that anything that looks like a line segment is a line segment.

In other words, if something looks straight, it is in this lesson.

Let's take a look at an example of a problem together.

Now let's find the value of X in this diagram.

It contains an isosceles triangle, which we can see because it has a hash mark on two of its sides and then we can see that there's an angle coming off one of the vertices, which is an exterior angle at that point, and that angle is 62 degrees.

How could we find the value of X and which angle facts would help us solve this problem along the way? It may require a couple of steps in order to get to our final solution and we may need to find some other angles within that diagram before we can find the value of x.

Pause the video while you think about how we might approach this and then press play when you're ready to continue.

Let's take a look at this together.

Now we could start with the fact that we have that 62 degree angle on a straight line and angles forming a straight line sum to 180 degrees.

So we could find that angle on the inside of that triangle by doing 180, subtract 62 to get 118 degrees.

Now we've done that, we can focus more so on the triangle itself, it's an isosceles triangle, which means that base angles in isosceles triangle are equal.

In other words, the X angle and the other angle, we don't know, they're equal to each other.

We also know that angles in a triangle sum to 180 degrees, so we could do 180, subtract the angle we know, 118, and then divide it by two to get 31 as our answer for the missing angle.

And that means our answer for X is also 31.

Problems can be made more complex by including more shapes.

For example, here we can see the figure from the previous question, but with an additional quadrilateral attached to it, we can also see that there are a pair of arrows that indicates that the top side of the quadrilateral is parallel to the bottom side of the isosceles triangle.

This is important because it may affect which angle facts we can use.

Let's find the value of Y in this diagram.

Pause the video, while you think about how we might approach this and which angle facts might be useful this time.

Are there any other angles you could find before you can find a value of y? Pause video while you think about it and press play when you're ready to continue.

Well, let's think about this together.

Now the angle which is marked Y is part of a quadrilateral and we know that angles in a quadrilateral sum to 360 degrees, but we can't use that fact straight away because there is another unknown angle in that quadrilateral.

If we can work out that unknown angle in the top left corner, then we could use the fact about angles in a quadrilateral to work out the value of why.

So how can we work out that unknown angle in the top left corner? Well, we know that angles in parallel lines which are alternate are equal and can we see which angle is alternate to the one we just highlighted? The angle in the bottom right corner of the isosceles triangle, which is 31 degrees, is alternate to the one that we're trying to find here in the top left corner, and they are between parallel lines, so that means they must be equal.

The angle is 31 degrees.

So now we know that now we can use the fact that angles are quadrilateral sum to 360 degrees.

We can make an equation by summing together all the angles that we know as well as the Y put it equal to 360.

We could then simplify that equation by adding together the angles we know and then we could subtract 235 from both sides to get Y equals 125.

Let's check what we've learned.

Here, we have a diagram containing an isosceles triangle.

Please find the value of A and justify your answer with reasoning.

Write down which angle facts you use along the way.

Pause video while you do this and press play when you're ready for an answer, A is equal to 66.

We can get to it by first using the fact that angles form in a straight line sum to 180 degrees and then we can do 180, subtract 114 to get 66.

Let's make this problem a bit more complicated now.

Let's find the value of B.

Justify your answer with reasoning.

Pause why you do it and press play when you're ready for an answer.

Answer is 66.

The base angles of an isosceles triangle are equal so we know that the angle labelled B is equal to angle labelled 66 degrees.

Let's now find another angle in this diagram.

Let's find this one.

What is the value of C? Justify your answer with reasoning.

Pause the video while you do it and press play when you're ready for an answer.

The answer is 48.

We can use the fact that angles in a triangle sum to what 180 degrees and then we can do C is equal to 180, subtract two lots of 66, which gives 48.

Let's now find another angle by making this question a little bit more complex, like so.

What is the value of D? Justify your answer with reasoning.

Pause the video while you do this and then press play when you're ready for an answer.

Answer is 48.

Alternate angles in parallel lines are equal and that's what we can see with this diagram here.

That top line we just added is parallel to the bottom line of the bottom side of the isosceles triangle.

Let's make this diagram a bit more complex again by joining it up to make a quadrilateral.

What is the value of E now? Justify your answer with reasoning.

Pause the video while you do this and press play when you are ready for an answer.

The answer is 107.

Angles in a quadrilateral sum to 360 degrees so we can do 360, subtract the angles that we know.

Now, you've just done five short questions, but this question could have been posed as it is now where we need to find the value of E and all those other unknown angles that you have previously found could be missing.

The way you solve that problem is to break it down the way we've just done, work out one angle and then use that to work out another angle.

Then use that to work out another angle and write down which facts to use along the way.

Well done.

Okay, it's over to you now for task A.

This task contains one question and here it is, find the value of each unknown represented by a letter and justify your answers with reasoning.

In other words, don't just find out the value.

Also write down which angle facts you use along the way.

Pause the video while you do this and press play when you're ready for answers, Let's go through some answers.

Question one, we defined A, B, and C here.

Well, A is equal to 53 because angles that form a straight line sum to 180 degrees.

B is equal to 53 as well, because base angles in an isosceles triangle are equal and C is equal to 74 because angles in a triangle sum to 180 degrees and we done 180, subtract two lots of 53.

And then we have D is equal to 51 here because base angles in an isosceles triangle are equal.

E is equal to 78 because angles in a triangle sum to 180 degrees.

So we subtract the angles we know from 180.

F is 102 because angles that form a straight line sum to 180 degrees, so we can do 180, subtract the 78 which just worked out and G is equal to H, which is equal to 39 because angles are triangle sum to 180 degrees and base angles in isosceles triangle are equal, which is why G is equal to H.

And then we have this one.

There's only one unknown here, but we can't work it out straight away.

We need to work out some other angles.

This angle is 62 degrees because it's vertical opposite the other angle and vertical opposite angles are equal.

This angle is 51 degrees because alternate angles in power lines are equal and it's alternate to the 51 at the top and now we can work out I, I is 67, because angles are triangles sum to 180 degrees.

And in this question we need to work out the value of J, but we need to work out other angles before we can get to it.

These two angles are 53 degrees because angles in a triangle sum to 180 degrees and then base angles in isosceles triangle are equal.

This angle is 127 degrees because angles form a straight line sum to 180 and 180 subtract 53 gives you 127, and now we have enough information to work out the value of J.

J is equal to 132 because angles in a quadrilateral sum to 360 degrees.

So far so good.

Now let's move on to the next part of this lesson where we're gonna look at some more complex problems. Some angle problems may provide you with information about the relationships between sizes of different angles within the figure.

For example, here we have two angles that are on a straight line, but we don't know how big one angle is is compared to the other.

These relationships may be multiplicative relationships and they may be expressed in the following ways.

They may be expressed as a multiplication.

We may be told explicitly that one angle is equal to two times the other angle or it may be expressed as a ratio.

We may be told that the ratio between the angles is equal to two to one, that says the same thing.

It still says that one angle is two times the other angle or may be expressed using algebra.

One angle may be labelled X and of all may be labelled 2X.

It means the same thing.

Again, that one angle is two times the other angle.

In each of these cases, the same equation can be formed.

We know that one angle is equal to double the other angle and we know that these two angles sum to 180 degrees, so we can write the equation X plus 2X equals 180.

Let's apply this now to a problem that looks a bit like this.

We have a Pentagon where we are given about one angle is 90 degrees, but we're not told what the other angles are.

However, we are provided with some information that tells us how big one angle is compared to another.

Let's use this information to find the size of angle B, C, D.

Pause the video while you think about what steps we might take and then press play when you're ready to look at this together.

Well, we could start by calling angle BCD X degrees and then we could use the information provided to us at the bottom of this diagram to express the other unknown angles in terms of X.

For example, we're told that angle EAB is equal to two times angle, B, C, D, so angle EAB must be 2X and we're told that angle ABC is equal to two times the angle we've just written down, angle EAB so angle ABC must be two times 2X, which is 4X and we're told that angle CDE is equal to angle EAB, so that means they both must be equal to 2X.

Now we've done that and we know that one of the angles is 90 degrees and the other angles are written in terms of X.

We can think about the fact that angles and append against sum two 540 degrees and that way we can use this to make an equation that looks a bit like this.

The sum of all those angles equals 540 degrees.

We can then simplify this equation, subtract 90 from both sides and divide each side by nine to get X equals 50.

And let's just double check which angle is we're looking for.

We're looking for angle BCD, which we have labelled just with X, not 2X or 4X.

So that means the size angle BCD is equal to 50 degrees.

Now this same question could have also been posed using ratios.

So rather than explicitly saying that one angle is two times another angle, we could write it like this.

The ratio between those three angles is two to four to one.

It still tells the same thing.

We can still see the angle EAB is two times angle BCD, because we have a two on the first part of the ratio and a one on the third part of the ratio.

And we can use all those same relationships by labelling the angle we want at the start X and then writing all the other ones as 2X and 2X and 4X to get to the same point we had earlier, where we can create an equation and find the angle we want.

Here we have another problem where we have a pair of parallel lines and isosceles triangle in between them.

We're not at told the size of any angles, but we are told that the ratio for angle EFD to angle DBC to angle BFG is equal to one to two to six.

Let's use this information to find the value of angle EFD.

Pause the video while you think about what steps we might take in order to solve this problem.

Even if the only thing you can think about at this point is how we might get started and then we'll work through it together in a second.

Pause while you think and press play to continue.

Okay, let's start by labelling angle EFD as X degrees and then we can use that ratio to write the other two angles that are part of that ratio in terms of X, like so.

We have angle DBC is equal to 2X and angle BFG is equal to 6X.

And then we need to think about how we can use angle facts in order to write other angles in terms of X until we get to a point where we have a relationship that sums to something or where one thing is equal to another thing and then we can use that to find the value of X itself.

Hmm, Let's see what we can do.

We have a point that is in between these two parallel lines and sometimes when we see this, we have a point in between two parallel lines and it's joined by a zigzag like shape, C to B to D to F to E.

It can be helpful to draw an additional parallel line like so.

And then we can think about how alternate angles in parallel lines are equal, because we have two pairs of alternate angles here.

We have these two pairs here.

One of those angles which is below our new line is alternate to the X and the angle's above that additional line is alternate to the 2X, so they must be equal.

That means that angle altogether in that isosceles triangle is 3X degrees and we know that base angles in isosceles triangle are equal.

So that angle sums to 3X degrees, so does the other angle.

And now we have three angles at a point on a straight line x, 3X and 6X.

Can you think about what fact we can use next? Angles form in a straight line sum to 180 degrees.

So we can write this as an equation where we've got the sum of those, which is 10X equals 180, and then we can divide both sides by 10 to get X equals 18.

And let's check what angle we're trying to work out.

We're trying to work out angle E, F, D and that one is labelled as X, not 2X or 3X, so that means that angle EFD is equal to 18 degrees.

Let's check what we've learned by doing a problem together now and we'll go break this problem down into lots of smaller problems. What do the interior angles of this polygon sum to? Pause while you write it down and press play when you're ready for an answer.

The answer is 720 degrees.

It's got six sides, it's a hexagon.

So now you're told this information.

Angle FAB is equal to 2X.

Angle CDE is equal to 1.

5 times angle FAB, and angle ABC is equal to two times angle FAB.

And what you need to do is use this information to write an expression in terms of X for angle CDE, and an expression in terms of X for the size of angle ABC.

Pause the video while you do that and press play when you're ready for answers.

Your answers are 3X for angle CDE, because that's 1.

5 times two and then you've got 4X for angle ABC because that's two times two and we have X as well.

So now you have all this information, find the value of x.

Pause the video while you do this and press play when you're ready for an answer.

The answer is 50.

You can get to it by making an equation based on adding these angles together and then you can simplify the equation to get X equals 50.

Now let's find an actual angle.

You know that X is 50, find the size of each of these angles, pause video while you do this and press play when you're ready for answers.

Angle FAB is 100 degrees.

That is by doing two times 50 angle.

ABC is 200 degrees.

That's by doing four times 50.

And angle CDE is 150 degrees.

That's by doing three times 50.

Okay, it's over to now for task B.

This task contains four questions and here are questions one and two.

Pause video while you do these and press play when you're ready for questions three and four.

And here are questions three and four.

Pause the video while you do these and press play when you are ready for answers.

Let's go through some answers.

Question one, you're given this information and we find the value of X.

Well, we could use information to write the other angles in terms of X and then we can use the fact that these angles sum to 720 degrees to make this equation here.

We could simplify the equation, rearrange it, and solve to get X equals 40.

And then question two, we've got a parallelogram with some line segments inside it and we need to find the value of X.

Let's start by writing it down as many other angles in terms of X as we can until we get to the point where we can create an equation.

The angle that is in the opposite diagonal corner to 4X is also equal to 4X degrees.

That's angle DEA, because diagonal opposite angles in a parallelogram are equal, that 4X is inside an isosceles triangle.

So we can work out an expression for the other two angles of isosceles triangle by subtracting 4X from 180 and then dividing by two to get 90, subtract 2X degrees for each of those angles.

We can then work out this angle here, CDA, by using the fact that angles on a straight line sum to 180 degrees when they are adjacent to each other.

That way we can do 180, subtract one of the angles we just worked out to get 90 plus 2X.

That angle is also inside an isosceles triangle, so we can use the same process again to work out the other two missing angles.

In the isosceles triangle we can do 180, subtract the angle we've just worked out and divide it by two to get 45 minus X degrees.

And then let's look at the angle in the bottom right corner, angle BCD.

We could write an expression for the entire that angle by summing together the two parts we've worked out and that'll give 3X plus 40 for the entire that angle.

Now that angle BCD is co-interior to the angle which is labelled 4X at the start.

Co-interior angles between parallel lines sum to 180 degrees.

So we can make this equation 4X plus 3X plus 40 equals 180.

We could simplify the equation and then rearrange it to solve the value of X as 20.

And then question three, we have a parallelogram and we have two angles given to us as a ratio.

However, neither of those two angles are equal to angle labelled X.

So what we may need to do here is use another letter to help us work out some other angles first.

For example, we could use one unit of this ratio and call it Y, which means that angle DEF would be equal to two Y degrees and angle BDE would be equal to three Y degrees.

We could use this fact and that co-interior angles in parallel lines sum to 180 degrees to create this equation and solve to get the value of Y as 36.

Now we know that Y is 36, we could work out some of those angles.

For example, the angle on the bottom right corner angle DEF is two y.

That would be two times 36 as 72.

And we could use that to find other angles in this diagram because some of the angles are equal to that one.

For example, in the top left corner of this parallelogram, that angle is diagonally opposite the one, the bottom right corner, so it must be equal.

That one must be 72 degrees as well.

And as part of an isosceles triangle, which means the other angle in the ISO lee triangle is also 72 degrees, and because we have a third angle in isosceles triangle, we could work that out to be 36 degrees and then we have X is on a straight line with the 36 degrees.

So we could do the fact that they add up to 180 degrees to find the value of X.

Then we have question four.

We have a pair of parallel lines with an isosceles triangle between them.

Let's start by drawing an extra line, which is parallel to the other two parallel lines.

We can use alternate angles between parallel lines to write an expression for each of these.

One is X degrees, the other is 2X degrees.

The entire of that angle is equal to 3X degrees and it's isosceles, which means the angle in the top left corner of this isosceles triangle is equal also to 3X degrees.

And then we could find the third angle of this isosceles triangle by using the fact that all three of those angles was sum to 180 degrees.

So we could do 180, subtract the sum of 3X, 2X and X to get an expression 180, subtract 6X degrees.

Now all three angles together on that straight line will sum to 180 degrees.

100 plus the angle we just worked out, 180 subtract 6X plus X will make 180 degrees.

We could simplify this equation and then we could rearrange it to solve it to X equals 20.

There are other ways we could have done that as well.

We could have used co-interior angles with the a hundred degrees to create an equation at the top on that top parallel line instead.

And sure you can probably find other ways as well.

Great work today.

Now let's summarise what we've learned during this lesson.

Problems can be solved using a variety of angle facts.

These may include facts about angles in parallel lines, interior and exterior angles and polygons, and many more.

Problems can be solved and the solutions justified using these angle facts.

Sometimes though, you might not be able to find the angle you want straight away and you may need to find other angles first.

That's okay, we can do that.

And finding unknown angles can make it easier to find other known unknown angles.

So the more angles you find, the easier it is it can be to find the answer you want.

Well done today.

Have a great day.