Polygons are multi-sided shapes with different properties. Shapes have symmetrical properties and some can tessellate. There are 360° in a full turn, 180° in a half turn and 90° in a quarter turn. A quarter turn is called a right angle. Angles around a point add up to 360°. This fact can be used to calculate missing angles. Angles on a straight line add up to 180°. This fact can also be used to calculate angles.
Okay, we're going to be looking at angles, basics of angles. Angles and triangles. And angles in quadrilaterals. Angles are very important in all sorts of endeavours construction. If you're constructing, building or furniture, you've got to make sure that all the pieces line up.
Angles are very important for that. So let's first off look at the names of angles here before we move on to angles and straight line angles around the point and so on. First thing to note is that the angles around a point are 360 degrees. That's by definition it could be anything. It could be 100,000.
We've chosen 360 because the numbers are just a bit neater. So what happens if we half that rather than go angles all the way around the point? If we do angles just halfway along angles on a straight line, well, we get 180 degrees, half of 360. We're half of all the way around 180 degrees. For the angles in a straight line, half of that would be 90 degrees.
And that's the angles in a right angle. If we're going to go smaller than that, say for example, 32 degrees, that's called an acute angle. And if we've got an angle that's larger than 90 but less than a straight line, that's an obtuse angle. Say 120 degrees there. And the last one after the one, people will forget it's all the way around past the straight line, but not quite all the way around the circle.
For example, 270 degrees. That's a reflex angle. More than a straight line, less than a full circle.
Okay, let's have a look at an application of angles on a straight line. We have three angles. There two of them we know one is an unknown and we need to see what the unknown angle X is. We know that 70 plus 60 is 130. And we know from our properties of straight lines that all three of these angles together must be 180.
So what we'd say is that X is equal to 180 degrees. Take away the other two added together. So that means that X is equal to the straight line, 180, take away 130. And so we get X is equal to 50 degrees in this case.
Okay, let's look at the angles around a point. Now we know that all four of these angles must add up to 360 degrees. And we're going to find that missing angle X again. So this time X is equal to 360 degrees. Take away the other three added up together.
So 100 plus 120 plus 25. And so we get that X is equal to 360 degrees. Take away 345 degrees. And so X is 15 degrees in this question.
Okay, looking at triangles. Now, before we move on to angles in a triangle, just some brief introduction to the names of triangles. We have an isosceles triangle here. These two marks tell us that these two sides are the same length. And so there's a line of symmetry up and down here that you can sort of see there.
I'm going to do that again.
OK, moving on to triangles. Now, just a couple of things about the names of no, not like that.
Okay, before we move on to angles in a triangle, we're just going to have a quick look at the names of some special triangles. This triangle on the left here, we've got two ticks on these lines, which means these lines, these two are both the same length, which implies it's an isosceles triangle. And these two angles are the same. By those two being the same length, you can see there's a sort of line of symmetry here, like a mirror reflection. That's the isosceles triangle, the equal axial triangle.
All three lengths are the same. And so these three angles will therefore be the same angle. And so if we were to calculate these angles, we know that it's 180 divided by three of the same angle. Each of these would be 60 degrees in the equal actual triangle.
Okay? Now to calculate the missing angle X in this triangle, we know that all three angles add up to 180 degrees. So once before, just like the example of the straight line, we say that it's 180 degrees. Take away the other two angles added together. And just like with our example of the straight line, we have 180 take away 130.
And so X in this case is 50 degrees. The angles on a straight line are the same as the angles in any triangle, 180 degrees.
Okay, now before we move on to angles in a quadrilateral, let's have a look at some properties of quadrilateral which is important to know. Now take a look at this quadrilateral at the top. This is the parallelogram. These lines tell you something about it. We know that these little ticks here, that these two parts of this line are the same.
In other words, this diagonal from bottom left to top right, this splits this line, the top left to bottom right line in half. Exactly. That means it bisects the line. So these diagonals bisect each other. This line is split half and half, as is this one.
So in the parallelogram, the diagonals bisect each other. The opposite sides are the same length. Red and red here, left and right, blue and blue. They're also parallel. These two blue lines are parallel and the red lines are parallel.
Hence parallelogram and the opposite angles are the same. You see these two red ones here and these two blue ones. So if you pause this diagram on YouTube, you can then remember all the properties of the parallelogram. If we move from the parallelogram and we impose a condition that rather than these arbitrary angles that these two bisectors meet at, they meet at 90 degrees. That automatically turns this parallelogram into a rectangle.
A few moments later, you can see that the angles here are also all 90 degrees. So if we make these angles 90 degrees, it means that these bisectors cut each other at 90 degrees as well. So that's how you go from the parallelogram to the rectangle. But notice in the rectangle the opposite sides, top and bottom are the same length and left and right are the same length, parallel to each other as well. But the top and the left aren't necessarily the same length.
That's what defines a rectangle moving down into the square. Now, if I impose another condition that the left and the top length are the same. Now all the sides are the same, they cut all right angles in the middle and each of the four angles is a right angle. Moving up to the parallelogram, again, we can move down to the Rhombus by imposing a different condition. Now we impose that all the sides are the same length.
We're still going to allow these diagonal bisectors to cut at different angles, not necessarily 90 degrees. But as soon as we decide that all lengths are the same, we now have a Rhombus. This does cut at 90 degrees here in the middle, these bisectors cut each other at 90 degrees. Opposite angles, red and red here are the same. Left, right, blue and blue are the same.
We have two pairs of equal angles. Just like with the parallelogram, we have two pairs of opposite equal angles. But now the only difference is all of the lengths are the same. And if we were to go from the bar back to the square, we would just be imposing that these angles are all right angles and then we get down to the square. Finally, if we want to go from the rhombus to the kite and remember the properties of the kite, that way we just decide that two of these lines are the same, but that the bi sectors meet at 90 degrees.
And these bisectors, unlike in the case with the parallelogram, they are not the same length. You see, this one up and down here is longer than left and right. So a kite has two adjacent sides that are the same length in blue here and the other two adjacent sides which are the same length in red and opposite angles, left and right here are the same up and down, though these opposite angles are not the same. Whereas in the parallelogram both top left to bottom right and bottom left to top right, both pairs of opposite angles are the same. Not the same in a kind.
So those are the properties and a quick way to remember how to go from parallelogram to rectangle to square or parallelogram to Rhombus to square and then from the Rhombus to the kite. And this diagram will tell you everything you need to know about the properties of these quadrilaterals.
Okay, now onto angles in a quadrilateral. Here we have an arbitrary quadrilateral, nothing special about it. We have three different angles, and we need to find the missing one, X. So, just as before, we know the property of quadrilaterals is that all of the angles in a quadrilateral, they add up to 360 in a quadrilateral, something special about a quadrilateral similar to a circle, the way around the point 360. And so we say that 360 degrees in this quadrilateral take away all three of these other angles added together.
And so we get 120 plus 60 is 180, plus 50 is 230. And so we get 360, takeaway 230. And X, therefore, is equal to 130 degrees in this example.
Moving on to an interesting exam question here. It says that this triangle ABC is an isosceles triangle. Work out the size of the angle marked X. Now, we know it's an isosceles as well, even if it didn't say it in the question, because these two ticks means that these two lines are the same sort of triangle we saw before, only this has been rotated 90 degrees. And if you recall, the two angles here are going to be the same.
So, as always in these questions, just put down what you do know. In this case, we know that this is 65 degrees. Up here, we have a missing angle, X. But we know that in a triangle, all the angles out to 180 degrees. So in this case, X is equal to 180 take away 65 to 65, which is 180 take away 130, leaving us with X being 50 degrees.
And finally, a more challenging question here for those of you who like a challenge. If you would care to put any of your answers in the comments section, my name is Adam Tudor for Sherpa. I'll be able to see if they're correct. Feel free to discuss it amongst yourselves. If you would require the answer, then feel free to email me or any other tutor at sherpa and they will be able to answer the question for you.
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