 ###### Devorah Goldblatt

Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT

Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.

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# Math Content Review III

Devorah Goldblatt ###### Devorah Goldblatt

Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT

Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.

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Plane geometry, what you need to know. First Let's take a look at this game I'm playing, I'm playing darts right now and I'm actually really close to winning but to win I need to get six points or more in my next throw and now I'm curious, what are my odds, what are my chances of hitting somewhere within this inner circle that will give me six or more points in my next throw and actually you can solve this using geometry. So in this episode, we're going to first talk about plane geometry; things like circles you know triangles, squares and then we'll move on to coordinate geometry and talk about things like slopes and lines. Let's get started.
Let's take a look at the top plane geometry topics. Now you can see there are a lot of topics here but there's a lot of plane geometry on the ACT, there are 14 of these out of 60. So nearly a quarter of the test is plane geometry. We're going to go over all of these concepts because I want you to feel really really good about them before we head into the test. We're going to talk about lines and angles, triangles, circles, squares and rectangles and we're going to do a multiple figure problem, a problem that combines two or more shapes.
On to lines and angles, there's just a couple of things that I want you to remember. Now I'll just drive this out really quickly. Angles along a line add up to 180 degrees, you'll probably remember this from math class but just in case you know anytime you got your line, you've got two or more angles on it, these two angles here are going to add up to 180 so for example if this was 120, this angle here would be 60. Okay another thing; angles around a point add up to 360 degrees and vertical angles are congruent. So Let's say I had angles around a point, here's my point right here, so here we got angles all around a point, four angles all the angular measurements added together would equal to 360 degrees and by the way, vertical angles are congruent. Vertical angles are these angles on the opposite sides for example this is a vertical angle to this, so these angles will be equivalent to each other, also this angle is equivalent to this angle because it's also a vertical angle.
On to one of my favorite topics with lines and angles; when two parallel lines are cut by a transversal, some really neat things happen. You're probably thinking transversal, Oh my God, what is she talking about? All transversal is, is just a line cutting through two parallel lines, let me draw a picture and now you're going to say "Oh, you know what Devorah I've seen that before" Do you remember this? It's you know, you'll usually see oh this is line one, oh this is line two, they're parallel and you've got a line that goes through them. When you see something like this, there's some really cool things you can infer about the angles. As long as you have one angle measurement, you can find all of them. So for example Let's say I told you, this angle here is 60 degrees, great okay Let's find all the other angles. Well, we know because this is 60 that this has to be 120 because they're along the line like we talked about. Great and now we know that this one has to be 60 as well because it's vertical and this one has to be 120 as well because it's a vertical and also because they're along the line with the other angles. Okay so this 60 here, this is 120 here. Okay but what about this bottom part? How can you figure out what those are? This is where another rule comes in, when you have two parallel lines cut by a transversal, alternate, interior angles are congruent. Those are the angles that I say you know, are kind of inside the Z that you see right here so this 60 degree angle right here is going to be equivalent to this angle right here, you see these opposite internal angles to these two parallel lines and the same thing here because this angle here is 120, this angle here opposite interior is going to be 120, so this is 120 and then this one here is going to be 60. Okay so hopefully you can see that we've got 120 and 60 here and 60 and 120 here and then once you've got this you know again this is a vertical so this is also 120 and this is a vertical so it's also 60 and there we go. So just from knowing one angle when you have two parallel lines cut by a transversal, you can find all the angles.
Let's move on to our next big topic, triangles. You'll usually see three or four triangle problems on the ACT so you want to feel pretty comfortable with triangles. Let's have a look at the first thing that they'll ask; area of the triangle, you need to know this formula. Remember on the ACT, they don't give you any formulas, you really need to know them. Area of a triangle is base times height divided by two. Let me give you some examples of how to find an area. The easiest ones are right triangles, Let's say I'm going to draw a right triangle here, there we go I'll put [in my little right angle and Let's say I tell you, okay you know this is four, this is five, okay easy to find the area. We've got our base, we've got our height, if you're ever not sure what's the base, what's the height, the base is at the bottom, the rule is that to be a height, you have to hit the base at 90 degrees so that's the definition of a height; it's hitting the base at 90. Here we go so that's easy, you got your height and then your base and you know the area will be 4 times 5 which is 20 over 2 which is 10; so area would be 20, 4 times 5 over 2 which would be 10. Pretty straight forward but, what happens when you don't see a height that stands out to you, that's hitting at a 90 degrees. What if you see a triangle instead that looks something like this and you look at it and you're thinking what's the height, what's dropping down? Hopefully you'll remember this from school, this is when you will drop an altitude, right so you will see this kind of dotted like almost fake line that you're drawing just to find the area dropping down from the bottom and now this is your base, this whole bottom and this is your height and on the ACT, they'll always be a way for you to figure out what that length is going to be so that you can find the area. So again these are the two major different permutations of how you're going to look for area. Always look for your base and your height, if it's not apparent because it's not a right triangle, make sure you're looking for a place you can drop an altitude and then you'll have your height and your base and once again it's going to be just whatever the base is times the height divided by two.
Let's move on, more about triangles, perimeter we know perimeter is everything added up, all the sides so it will be all three sides added together is the perimeter of the triangle. All three angles of a triangle add up to 180 degrees, that's another thing that is tested occasionally with triangles. So all your three angles adding up to 180. On to another common triangle concept, similar triangles. Similar triangles have equal angles and proportional sides, I like to say they're kind of like Doctor Evil and Mini me. One of the triangles will kind of be a miniature version of the second one, so let me show you what I mean; you'll see something like this, you'll have a big triangle and a little triangle and then you'll know that the angles are congruent, maybe they'll just be lines like these [IB] you know oh, this angle's the same as this one and this angle is the same as this one which means that the third angles also have to be congruent. In a case like this, you know okay if the angles are the same, the size have to be in proportion. So Let's say you know, that this one has a side of two and this one has a side of four. Proportion. I know that all these sides are going to be double, all the sides of this triangle so for example this one's you know two, three, four, this would be something like four, six and eight. So you see the sides are proportional and again we talked about proportions in the pre-algebra episode, there will be times in the ACT where you're going to actually have to find one of the missing sides using proportions and for more practice just head to your bonus materials, I've got some [IB] to practice this.
Let's move on to right triangles, this is the most common triangle you'll see on the ACT. So for right triangles, a few things you need to know. First one angle is equal to 90 degrees that's a right triangle, second you got the Pythagorean Theorem to know; a squared plus b squared equals c squared, you need to know this so memorize it if you don't know it by heart already. Let me just show you a quick example of using the Pythagorean Theorem; if I had a right triangle right here and I knew for example that this side was three and this side was four and I was looking for the hypotenuse, this side opposite my right angle and we don't know what it is. Okay we know that 3 squared plus 4 squared is going to give us X squared so we have 3 squared plus four squared equals X squared. Okay, so we know that nine plus 16 is X squared. So 25 equals X squared and so therefore you are going to take the square root of both sides to remove your exponent here and X is the square root of 25 which is equal to five. So here we go, our third side here would be five on our right triangle. That's an example of using the Pythagorean Theorem. And we'll have another example later to practice again.
Some other things with right triangles, 45-45-90 and 30-60-90 triangles are the most commonly tested kind of right triangles. What this means is that you'll see triangles that either have angles that are 45 degrees, 45 degrees and 90 or 30, 60 and 90. And when you see triangles like these they have very specific side of proportions always. So let's start with the 45-45-90 and I'll show you what I mean. So if I have got a 45-45-90 triangle here, okay so we know 45-45-90, there's some specifics that are always the same. Always you'll have the sides in the following ratio: you're going to have X-X-X square root of two and you're thinking, what is she talking about? I'll show you. What this means is that the side opposite the 45 degree angles, we'll call those X, the short sides. So we are going to have these we call those X. The side opposite the 90 degree angle, your hypotenuse, is going to be radical two times the shortest side. Well here we've got two short sides so radical two times either of the sides always. So that's your proportion, 45-45-90, you've got X if you're going to call the side opposite of 45 X; you've got X-X-X radical two.
Let me put it into practice so you see what I'm talking about. If I told you for example, one of the sides was say three. Well, it's a side opposite of 45 degree angle, that's our X, that's our shortest side for the proportion so we know from this proportion that always the side opposite the 90 degrees is radical two times the shortest length. So now I know even if I'm doing the Pythagorean Theorem, that this hypotenuse right here, the side opposite the 90 degree angle is radical two times three so three radical two and also because these are both opposite 45 degree angles, these are equivalent so both of these sides are going to be three. So that's how side proportions will work, on 45-45-90.