Lesson video

Hello, my name is Mr Clasper, and today we're going to look at finding missing angles inside a triangle.

In this lesson, we're going to look at the angles inside a triangle.

The diagram shows a triangle with three highlighted angles.

What we could do is we could rearrange the angles inside the triangle and place them on a straight line.

So if we take the red angle, followed by the blue angle, followed by the green angle, we can see that they fit onto a straight line.

And for prior learning, we should know that angles on a straight line have an angle sum of 180 degrees.

Therefore, the angles inside a triangle also have an angle sum of 180 degrees.

This can be useful when we solve problems with many angles.

Let's take a look at this example.

So we know that all three of my angles inside my triangle must have a sum of 180.

Therefore, I can write this equation.

I can also add my two known angles together, which leaves me with 137 degrees plus 'a', must be equal to 180 degrees.

And if I subtract 137, that means that the value of 'a' must be 43 degrees.

Let's have a look at this example, we need to calculate the size of angle 'a'.

This one's a little bit more tricky, as there is another angle which we need to find first.

To do this, we need to look at our 130 degree angle.

And this angle actually resides on a straight line, at the base of the triangle.

So from this we should know that 130 plus something must be equal to 180, because angles on a straight line have a sum of 180 degrees.

That means that our missing interior angle of the triangle must be 50 degrees.

From this point, we can work as we did in the previous example.

So we know that our three angles inside our triangle must have a sum of 180 degrees.

And simplifying this, we are left with 110 degrees plus 'a' must be equal to 180.

And subtracting 110 would leave us with the solution 'a' is equal to 70 degrees.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So remember that angles inside a triangle must have a sum of 180 degrees.

If we look at question two, we can see that the first triangle has a square, this indicates a 90 degree angle.

So you're going to calculate with 90 degrees and 35 degrees, which should lead you to the conclusion that 'z' must be equal to 55.

And if we look at the second example, we need to find our interior angle first.

So 180 degrees, subtract 109 degrees, because they are on a straight line, would leave you with 71 degrees as an interior angle to our triangle, and if we subtract 71 degrees and 47 degrees from 180 degrees, we find our answer of 62 degrees.

For our next examples, we're going to be using isosceles triangles.

There are two important facts that need to be remembered by isosceles triangles.

They have two equal angles and two equal sides, indicated by the small lines on either side of our triangle.

Have a look at this triangle.

Which two angles are equal? The correct answer is 'a' and 'c'.

Angles 'a' and 'c' are equal, and our two equal sides are BA and BC.

Have a look at this triangle.

Which two angles are equal? The correct answer is 'b' and 'c'.

This is because my equal sides have now changed.

So my equal sides are AC and AB.

And this example, which two angles are equal? The correct answer is 'a' and 'b'.

This is because my two equal sides are CB, and CA Let's take a look at this problem, we need to calculate the size of angle 'a', but our problem is that, we have two unknown angles inside our triangle.

Given that this triangle is isosceles, indicated by the two small lines telling us that these two sides are equal in length, we should know that the other unknown angle is 70 degrees.

Now that I know that these two angles are 70 degrees, I can start my problem in a similar way to previous the examples.

So we know that the three angles inside the triangle must have a sum of 180 degrees.

Therefore, if I subtract 140 from 180 degrees, this would leave me with the conclusion that 'a' is equal to 40 degrees.

Let's take a look at this example.

In this isosceles triangle, our other unknown angle is equal to 'a'.

And we know that these three angles must have a sum of 180 degrees.

If I simplify this, I could write this as an equation, which will be 50 degrees plus '2a' is equal to 180 degrees.

And this must therefore mean that if I subtract 50 degrees from my 180 degrees, that two lots of 'a' are equal to 130 degrees.

I only need one of these two a's.

So if I divide both sides of my equation by two, I will get 'a' is equal to 65 degrees.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So remember, in isosceles triangle we have two equal angles.

So for the first example, 'a' was 68 degrees, as it is equal to the other given 68 degree angle.

And from here we can find out our other missing angle, which was 44 degrees.

For 'm', if we subtract 44 from 180, we would get 136.

And having this would give us 68 degrees, which gives us our value of 'm'.

And for the last example, the angle at the top is 90 degrees.

So we can calculate 180 minus 90 degrees, which means that our two remaining angles which are equal are also equal to 90 degrees and therefore one of these angles must be 45.

Let's take a look at this example.

We need to calculate the size of angle 'a'.

However, we have no angles inside our triangle.

But we do have an exterior angle of 110 degrees, as the exterior angle resides on a straight line, we can use our angle facts related to straight lines.

So angles on a straight line also have a sum of 180, meaning that our interior angle is 70 degrees.

As this is an isosceles triangle indicated by our two small lines on either sides, this means that our other angle on the base is also 70 degrees.

So this means that our three angles inside our triangle are 70 degrees, 70 degrees, and 'a', and they must have a sum of 180.

If I simplify this, this means that 140 degrees plus 'a' must be equal to 180 and therefore, 'a' must be 40 degrees.

Here's some questions for you to try.

Pause the video to complete your task and click resume once you're finished And here are your solutions.

So if we take a look at the bottom left, we have an equilateral triangle.

Equilateral triangles have three equal sides, which have a sum of 180 degrees.

This means that each of the three angles must be 60 degrees.

If the interior angle is 60 degrees, that means the exterior angle 'x' must be 120 degrees, as these both reside on a straight line.

And my example in the bottom right, I have an equilateral triangle.

We know that all three angles must be 60 degrees.

And if I look at my reflex angle, 'y', this must be 300 degrees as angles around the point must have a sum of 360 degrees.

So 360 degrees minus my 60 degree angle inside the equilateral triangle, would leave me with 300 degrees.

Here's our last question.

Pause the video to complete your task and click resume once you're finished.

And here is your solution.

So for this question, it will be useful to work in steps of find out as many angles as you can that will help you get closer to your goal.

So if we have a look at the bottom isosceles triangle, we know that angles QRS, and QSR are are both equal as it is an isosceles triangle, so therefore, these both must be 44 degrees.

Once we know that these two angles are 44, we can use angles in a triangle having a sum of 180 degrees to work out the other missing angle which is 92 degrees.

If we have a look at the triangle, PQS, I have two of the three angles, so I can use my angles in a triangle sum again.

So, if I subtract 20 and 40 from 180 degrees this leaves me with 120 degrees and my final step Is to subtract 120 and 92 from 360 degrees as these all reside around a point and will leave me with 148 degrees as my size of angle 'x'.

And this brings us nicely to the end of our lesson.

I hope you've had fun finding missing angles inside triangles.

If you're feeling confident, why not have a go the exit quiz.

Take care and hopefully see you soon.