Do you remember what is special about the interior angles of a triangle? They always add up to 180 degrees. And whenever you use this fact, you need to give the following reason. If a triangle has no equal sides, we call it a scalene triangle. Please note, the angles are also all different in a scalene triangle. Do you remember what we call a triangle with two equal sides? It is called an isosceles triangle. This triangle also has two equal angles, which lie at the ends of the equal sides, or opposite the equal sides. Do you remember the name of the triangle with three equal sides? It is an equilateral triangle. This triangle also has three equal angles. And because the angles of a triangle always add up to 180 degrees, each angle is 60 degrees. And what type of triangle is this triangle? It is a right-angled triangle. This angle is 90 degrees, which is of course the right angle. and the side opposite this angle is called the hypotenuse. Please pause to study this important information. Do you remember what type of angle we get when we extend a side of a triangle? We get an exterior angle of the triangle. These two angles are called the opposite interior angles. An important fact to remember here is that the exterior angle is always equal to the sum of the opposite interior angles. And whenever you use this fact, you need to write the following reason. Please pause to study this information. Let's now use our knowledge of triangles. Together with vertically opposite angles, and parallel lines, to answer a few problems or riders. If you can't remember all of this, we suggest you first listen to the previous Grade 9 lessons, and maybe some Grade 8 lessons on this topic. Here is the first problem, or rider. Please pause to read it. To find x, we can use triangle ABC, together with its exterior angle, 2x. As we have said, 2x is an exterior angle. and this is equal to the sum of the opposite interior angles, 50 degrees, and x plus 10 degrees. And don't forget to write the reason in this step. To solve this equation, we begin by adding 50 degrees to 10 degrees on the right. We then subtract x from both sides. x is therefore equal to 60 degrees. The exterior angle is therefore 2 or To find y, we need to use this triangle. And because it is an isosceles triangle, angle CAD is also equal to y. Don't forget to write the reason in this step. We can therefore write that y plus y plus 120 degrees equals 180 degrees. Why? Because the sum of the interior angles of a triangle is always 180 degrees. To solve the equation, we begin by adding like terms on the left. We then subtract 120 degrees from both sides. and then we divide both sides by 2. y is therefore equal to 30 degrees. Please pause to check my working. Here is the next question. Please pause to read it. We begin with the parallel lines and the letter F. This means that angle DEC equals 90 degrees. Why? Because we have corresponding angles, and AB is parallel to DE, we can now use triangle DEC. Can you see how? We can use the fact that the angles of a triangle add up to 180 degrees. To solve the equation, we begin by collecting like terms on the left. We then subtract 80 degrees from both sides. And then we divide both sides by 5. x is therefore equal to 20 degrees. To find y, we again use parallel lines and the letter f. This means that y is equal to 3x minus 10 degrees. And don't forget to write the reason. We have just seen that x is 20 degrees, so we can substitute 20 degrees in place of x here. 3 times 20 degrees equals 60 degrees. And 60 degrees minus 10 degrees equals 50 degrees. So y is equal to 50 degrees. Please pause to check my working. Here is the next question. Please pause to read it. We begin by using the parallel lines and the letter Z or N. This means that 3X minus 24 degrees equals 2X. Why? Because there are alternate angles, and AB is parallel to DE. To solve the equation, we begin by subtracting 2X from both sides. We then add 24 degrees to both sides. X is therefore equal to 24 degrees. Before we can find Y, we first need to use these straight lines. And the fact that vertically opposite angles are equal, angle C1 is therefore equal to 57 degrees. We are now ready to use triangle ABC, because Y is the only unknown interior angle, We are given that angle A is equal to 3x minus 24 degrees. And we have just calculated that x is 24 degrees. So to find angle A, we need to substitute 24 degrees in place of x. Angle A is therefore equal to 48 degrees. This means we can write the following equation, because the angles of a triangle always add up to 180 degrees. To solve the equation, we begin by adding 48 degrees and 57 degrees on the left. We then subtract 105 degrees from both sides. y is therefore equal to 75 degrees. Please pause to check my working. Let's end with this question. Please pause to read it. To find x is easy, we just need to use triangle ABC. and angle C1 is of course 90 degrees. We can therefore write the following equation, and don't forget the reason. To solve the equation, we begin by adding 38 degrees and 90 degrees. We then subtract 128 degrees from both sides. X is therefore equal to 52 degrees. To find Y, you need to recognize that triangle BCD is an isosceles triangle. This means that angle C2 and Y are equal. Can you see a way of finding the size of angle C2? Let me help you. We need to use parallel lines, as well as the letter Z or N. Now can you see how to find angle C2? Yes, angle C2 is equal to 52 degrees, because alternate angles are equal, and AB is parallel to CD. This means that Y is also equal to 52 degrees. And the reason is, isosceles triangle BCD. Lastly, to find Z, we need to use the fact that we have a straight line here. This means that Z plus 52 degrees plus 52 degrees equals 180 degrees. And the reason is angles on a straight line, or just supplementary angles. To solve the equation, we begin by adding 52 degrees and 52 degrees on the left. We then subtract 104 degrees from both sides. Z is therefore equal to 76 degrees. This completes the lesson. Good luck with the test.