WEBVTT
00:00:02.710 --> 00:00:09.400
In this video, we’re going to look at a key fact about the angles in triangles and then use it to solve some problems in this area.
00:00:12.090 --> 00:00:15.090
So, the key fact that we need about the angles in triangles, is this.
00:00:15.320 --> 00:00:20.750
The sum of the interior angles in any triangle is 180 degrees.
00:00:22.170 --> 00:00:25.840
So, by interior angles, we mean those angles that are inside the triangle.
00:00:26.100 --> 00:00:29.500
So, the angles that are labelled 𝑎, 𝑏, and 𝑐 in the diagram here.
00:00:30.970 --> 00:00:34.110
Now, we’re going to look at one way of proving this fact.
00:00:35.480 --> 00:00:36.680
I’m gonna use the diagram here.
00:00:36.680 --> 00:00:42.590
And the first thing I’m going to do is I’m gonna add in a line that is parallel to the base of this triangle.
00:00:43.900 --> 00:00:47.610
So, this line that I’ve added at the top of the diagram, is parallel to the base.
00:00:47.610 --> 00:00:50.310
And you can see, I’ve marked that on using the arrows.
00:00:51.810 --> 00:00:55.650
Now, this proof relies on some key facts about angles in parallel lines.
00:00:55.810 --> 00:00:58.090
And that’s why I chose to add this extra line in.
00:00:59.870 --> 00:01:04.270
So, I’m going to think, first of all, about this angle here, that I’ve marked in red.
00:01:05.760 --> 00:01:12.190
Now, if you look carefully at the diagram, you should see that it has a particular relationship with angle 𝑎, this angle here.
00:01:12.540 --> 00:01:17.050
They’re what’s referred to as alternate interior angles in parallel lines.
00:01:18.260 --> 00:01:27.110
And if you recall a key fact about those, is that alternate interior angles are equal, which means that this angle here is the same as angle 𝑎.
00:01:27.140 --> 00:01:28.970
So, I can label it as 𝑎.
00:01:30.490 --> 00:01:36.240
So, if I write the reason down as well, it’s because, as we already said, alternate interior angles are equal.
00:01:37.870 --> 00:01:40.350
Now, let’s think about this angle here.
00:01:41.630 --> 00:01:48.180
And again, if you look carefully at the diagram, you’ll see that this angle, which I’ve marked in green, has a special relationship with angle 𝑏.
00:01:48.300 --> 00:01:49.600
And it’s the same relationship.
00:01:49.600 --> 00:01:53.810
They are also alternate interior angles in parallel lines.
00:01:55.390 --> 00:02:02.370
Now, given that alternate interior angles are equal, that means I can label this little angle here, I can give it the letter 𝑏.
00:02:03.760 --> 00:02:07.370
And that’s due to the same reasoning as before, just in a different part of the diagram.
00:02:09.000 --> 00:02:09.340
Right.
00:02:09.340 --> 00:02:12.950
Finally, if we think about this top part of the diagram, so this part here.
00:02:13.310 --> 00:02:17.860
I now have angles 𝑎, 𝑐, and 𝑏, all next to each other on a straight line.
00:02:19.160 --> 00:02:25.200
And again, if you recall a key fact about angles on a straight line, it’s that they add up to 180 degrees.
00:02:27.080 --> 00:02:35.920
So, what this tells me, is that 𝑎 plus 𝑏 plus 𝑐, I’ll put them in that order, is equal to 180 degrees.
00:02:37.370 --> 00:02:41.600
And the reason, remember, was that angles on a straight-line sum to 180 degrees.
00:02:42.980 --> 00:02:44.290
So, that’s what we were trying to show.
00:02:44.290 --> 00:02:54.850
We were trying to show that 𝑎, 𝑏 and 𝑐, which were the angles in the triangle, add to 180 degrees, by using facts about angles in parallel lines, specifically alternate interior angles.
00:02:55.050 --> 00:03:00.210
And using the fact that angles on a straight line sum to 180 degrees, we’ve proven this fact.
00:03:02.450 --> 00:03:05.950
Right, now, let’s look at how we can apply this fact to answering a couple of questions.
00:03:06.160 --> 00:03:10.430
So, we have a diagram and we’re asked to find the measure of angle 𝐵𝐴𝐶.
00:03:10.660 --> 00:03:16.930
So, that means the angle formed when we move from 𝐵 to 𝐴 to 𝐶, so it’s this angle here.
00:03:18.470 --> 00:03:19.970
So, that’s the angle we’re looking to find.
00:03:20.180 --> 00:03:24.990
But we can’t work it out straightaway because we only currently know one of the angles in the triangle.
00:03:25.500 --> 00:03:31.100
We can, however, work out what the other angle in the triangle is, so angle 𝐵𝐶𝐴.
00:03:32.300 --> 00:03:38.940
Because what you’ll notice is that angle 𝐵𝐶𝐴 is on a straight line with this angle of 163 degrees.
00:03:40.530 --> 00:03:45.710
So, we can use that fact about angles on a straight line, to work out angle 𝐵𝐶𝐴 first.
00:03:47.480 --> 00:03:54.930
So, the measure of angle 𝐵𝐶𝐴 is 180 minus 163, and that gives us 17 degrees.
00:03:55.050 --> 00:04:00.940
And the reasoning for that, which I’ve written at the side, is that angles on a straight line sum to 180 degrees.
00:04:02.800 --> 00:04:05.500
So, I can mark angle 𝐵𝐶𝐴 on to the diagram.
00:04:06.640 --> 00:04:07.490
And there it is.
00:04:07.910 --> 00:04:16.580
Now, we have enough information to work out this angle 𝐵𝐴𝐶 that I was originally asked for because, again, we know the angles in a triangle sum to 180 degrees.
00:04:16.700 --> 00:04:19.420
And if I know two of them, I can work out the third.
00:04:21.170 --> 00:04:26.260
So, to find angle 𝐵𝐴𝐶, we’re gonna do 180 minus 100 minus 17.
00:04:26.290 --> 00:04:29.100
That’s subtracting both of the other two angles in the triangle.
00:04:30.430 --> 00:04:34.370
And so, that gives us 63 degrees for the measure of angle 𝐵𝐴𝐶.
00:04:34.570 --> 00:04:39.060
And the reasoning, as we said, angles in a triangle sum to 180 degrees.
00:04:40.730 --> 00:04:45.360
So, often in questions like these, you can’t work out the angle you’re looking for immediately.
00:04:45.540 --> 00:04:51.900
You may have to work out other angles in the diagram first, by using facts about angles in a triangle or angles on a straight line.
00:04:52.040 --> 00:04:55.200
And once you’ve got those, you can then work out the angle you’re looking for.
00:04:57.750 --> 00:04:59.790
Okay, here’s the next problem we’re going to look at.
00:05:00.190 --> 00:05:01.650
We’re given a diagram of a triangle.
00:05:01.890 --> 00:05:03.970
And we’re asked to find the value of 𝑥.
00:05:04.070 --> 00:05:11.380
And if you look at the diagram, you’ll see that all three of the angles in this triangle are expressed in terms of this unknown variable 𝑥.
00:05:13.270 --> 00:05:17.800
So, thinking about how to approach this problem, we need the key fact about angles in a triangle.
00:05:19.130 --> 00:05:22.960
And of course, it’s this, that the angles in a triangle sum to 180 degrees.
00:05:23.730 --> 00:05:27.540
So, think about how you can use this fact to help you answer this problem.
00:05:29.000 --> 00:05:36.180
Well, we don’t know what the values of the angles are, in terms of their numeric measures, but we do know them in terms of this unknown letter 𝑥.
00:05:36.420 --> 00:05:38.510
So, we can start writing an equation.
00:05:40.010 --> 00:05:43.750
So, if I add up all of the different angles in this triangle.
00:05:45.130 --> 00:05:54.350
So, that will be 𝑥 plus two 𝑥 minus 10 plus 𝑥 plus 30, if I add them altogether, it has to give me 180 degrees.
00:05:55.960 --> 00:06:01.020
So, here is an equation, or the beginning of an equation, that I can use to work out this letter 𝑥.
00:06:02.950 --> 00:06:05.540
So, the next thing I’d want to do is to simplify this equation.
00:06:05.790 --> 00:06:11.620
If I look at the left-hand side, I’ve got 𝑥 plus two 𝑥 plus 𝑥, so I’ve got four 𝑥 overall.
00:06:13.080 --> 00:06:17.440
And then, I’ve got minus 10 plus 30, so overall, I’ve got plus 20.
00:06:18.940 --> 00:06:23.640
So, simplifying this equation, I have four 𝑥 plus 20 is equal to 180.
00:06:24.420 --> 00:06:25.950
Now, I want to solve this equation.
00:06:25.950 --> 00:06:30.360
So, the first step is gonna be to subtract 20 from both sides.
00:06:32.020 --> 00:06:34.950
And when I do, I get four 𝑥 is equal to 160.
00:06:35.550 --> 00:06:40.200
Next, to work out the value of 𝑥, I need to divide both sides of the equation by four.
00:06:41.660 --> 00:06:46.490
And when I do that, it gives me 𝑥 is equal to 40, which is therefore the answer to this question.
00:06:48.200 --> 00:06:59.430
So, this question involved using that fact about the angle sum in a triangle, but also some algebra skills, in terms of setting up and then solving an equation, in order to work out the value of this unknown letter 𝑥.
00:07:01.580 --> 00:07:02.910
Okay, the next question.
00:07:03.330 --> 00:07:07.230
The ratio of the three angles in a triangle is five to four to nine.
00:07:07.590 --> 00:07:09.860
Find the size of the smallest angle.
00:07:11.950 --> 00:07:15.510
So, with questions involving ratio, there are lots of different approaches that you can take.
00:07:15.710 --> 00:07:19.810
I’ll demonstrate two of these approaches, and then you could decide which of them you prefer.
00:07:21.710 --> 00:07:29.180
So, the first approach is an algebraic approach, where we say, well, we don’t know what these angles are, but we know they’re in the ratio of five to four to nine.
00:07:29.300 --> 00:07:38.750
Which means I could call these angles five 𝑥, four 𝑥, and nine 𝑥, where 𝑥 is just representing some unknown value.
00:07:38.960 --> 00:07:41.360
But this keeps the five to four to nine ratio.
00:07:43.030 --> 00:07:47.250
Now, remember our key fact, that the sum of the angles in a triangle is 180 degrees.
00:07:47.490 --> 00:07:49.250
So, I can turn this into an equation.
00:07:50.660 --> 00:07:55.610
So, if I add plus signs between those three terms, then it’s equal to 180.
00:07:55.860 --> 00:07:59.860
So, what I’ve done is set up an equation involving this unknown letter 𝑥.
00:08:01.690 --> 00:08:03.480
So, now, I can simplify this equation.
00:08:03.610 --> 00:08:07.050
Add five 𝑥 plus four 𝑥 plus nine 𝑥 becomes 18 𝑥.
00:08:08.270 --> 00:08:10.780
So, I have 18 𝑥 is equal to 180.
00:08:11.130 --> 00:08:18.960
To solve the equation then, I need to divide both sides of the equation by 18, and this gives me that 𝑥 is equal to 10.
00:08:19.750 --> 00:08:23.060
Now, the question, remember, said find the size of the smallest angle.
00:08:23.280 --> 00:08:28.570
So, the smallest angle is this four 𝑥, the one that has the least parts of this ratio.
00:08:29.130 --> 00:08:33.110
So, in order to work out the smallest angle, I need to multiply 𝑥 by four.
00:08:34.590 --> 00:08:35.530
So, I have four 𝑥.
00:08:35.530 --> 00:08:37.710
Four times 10 is equal to forty.
00:08:38.570 --> 00:08:43.230
And this gives me the answer to the problem, which is that the size of the smallest angle is 40 degrees.
00:08:44.910 --> 00:08:49.080
So, that’s one way of approaching it, treating it like an algebra problem and setting up an equation.
00:08:49.660 --> 00:08:54.660
The other way that I’d like to think about ratio problems, is thinking about the parts of the ratio.
00:08:55.050 --> 00:08:57.790
So, we have a ratio of five to four to nine.
00:08:58.120 --> 00:09:00.940
If I add them together, five plus four plus nine is 18.
00:09:01.120 --> 00:09:05.110
So, in total, there are 18 equal parts in this ratio.
00:09:07.010 --> 00:09:12.450
Now, as we’ve said many times in this video, the sum of the angles in a triangle is 180 degrees.
00:09:12.650 --> 00:09:16.400
So, those 18 parts together are worth 180.
00:09:17.860 --> 00:09:22.150
I want to work out the size of the smallest angle, so I want to know what four parts are worth.
00:09:22.390 --> 00:09:24.480
And there are lots of different ways I could do this.
00:09:24.610 --> 00:09:27.890
I could work out what one part is, by dividing by 18.
00:09:29.210 --> 00:09:31.330
So, that would give me one part is equal to 10.
00:09:31.840 --> 00:09:34.760
And then, to find four parts, I’d have to multiply by four.
00:09:36.090 --> 00:09:40.270
And so, of course, that gives me the same answer, as before, of forty degrees.
00:09:42.090 --> 00:09:44.940
I could, perhaps, have approached this ratio in a slightly different way.
00:09:44.940 --> 00:09:48.940
Instead of finding one part, I could’ve found, perhaps, two parts.
00:09:50.210 --> 00:09:56.440
So, that would’ve meant dividing both sides by nine, and then I’d just have doubled it to find the four parts as 40.
00:09:58.030 --> 00:10:10.230
So, whichever approach you prefer, either an algebraic approach involving setting up an equation or thinking about ratio in terms of equal parts and dividing down and then scaling back up to however many parts you’re looking for.
00:10:12.550 --> 00:10:14.510
Okay, the final question that we’re going to look at.
00:10:14.510 --> 00:10:19.080
It says, one angle in an isosceles triangle is 50 degrees.
00:10:19.350 --> 00:10:21.440
What could the other angles be?
00:10:23.240 --> 00:10:25.770
Now, there are a couple of words that jump out at me from that question.
00:10:25.980 --> 00:10:27.530
The first is isosceles.
00:10:27.750 --> 00:10:33.630
Now, remember, an isosceles triangle is a particular type of triangle which has two sides the same length.
00:10:33.900 --> 00:10:38.000
But also, in terms of angles, it has two angles that are equal to each other.
00:10:38.980 --> 00:10:41.890
The other word that jumps out at me is that word could.
00:10:43.300 --> 00:10:48.570
Because whenever I see that word, that suggests that there’s more than one possible answer to this question.
00:10:48.780 --> 00:10:51.320
So, I need to think carefully about why that might be the case.
00:10:53.270 --> 00:10:54.790
So, let’s have a go at this problem then.
00:10:54.910 --> 00:11:01.550
I’ve drawn an isosceles triangle, and I’ve indicated two sides as the same length by including this little line on them.
00:11:02.950 --> 00:11:06.280
Now, it tells me, one angle in the isosceles triangle is 50 degrees.
00:11:06.550 --> 00:11:12.040
So, it could be the case that this angle here is 50 degrees.
00:11:13.720 --> 00:11:23.170
Now, if that is the case, because isosceles triangles have two equal angles, and because they’re the base angles where these equal sides join the third side.
00:11:23.470 --> 00:11:29.760
That would mean that this angle here would also have to be 50 degrees because they’d have to be equal to each other.
00:11:31.990 --> 00:11:37.480
The third angle then, so this angle here, I would work out using my fact about angles in a triangle.
00:11:39.440 --> 00:11:51.330
So, if the angles in the triangle sum to 180 degrees, this angle here would be worked out by doing 180 minus 50 minus 50, so subtracting the other two angles.
00:11:52.650 --> 00:11:55.310
So, this would give me 80 degrees for the third angle.
00:11:56.650 --> 00:11:59.650
So, there is one possibility for the three angles in this triangle.
00:11:59.650 --> 00:12:03.630
They could be 50 degrees, 50 degrees, and 80 degrees.
00:12:05.250 --> 00:12:06.440
But the question said could.
00:12:06.440 --> 00:12:10.330
And as I said, that suggests that perhaps there is more than one answer.
00:12:10.530 --> 00:12:12.160
So, I’ll draw the diagram out again.
00:12:13.810 --> 00:12:23.360
So, perhaps instead of the 50 degrees being one of those two equal base angles, perhaps the 50 degrees is actually the third angle in the triangle.
00:12:23.640 --> 00:12:24.440
This angle here.
00:12:26.390 --> 00:12:28.090
And now, if I want to work out the other two.
00:12:28.630 --> 00:12:31.540
Well, the angles in a triangle sum to 180 degrees.
00:12:32.030 --> 00:12:36.640
If I subtract this 50 degrees, then I’ve got 130 degrees left.
00:12:36.850 --> 00:12:41.520
And as these two angles are both equal, they must both be half of that.
00:12:42.940 --> 00:12:47.650
So, they’re both equal to 180 subtract 50 and then divided by two.
00:12:48.810 --> 00:12:52.530
And so, that gives me 65 degrees for each of these angles.
00:12:53.940 --> 00:13:00.990
So, that gives me another possibility of 50 degrees, 65 degrees, 65 degrees for the angles in this triangle.
00:13:02.440 --> 00:13:04.610
So, that gives me my two possibilities.
00:13:06.550 --> 00:13:14.080
So, in summary then, in this video, we’ve looked at the key fact that the sum of the interior angles in any triangle is 180 degrees.
00:13:14.240 --> 00:13:19.560
We’ve seen how to prove it using facts about alternate interior angles in parallel lines.
00:13:19.750 --> 00:13:22.820
And then we’ve applied this fact to answer some problems.