If you are a student at a high school, college, university or you take an online course anywhere around the world, then you are probably familiar with geometry. Geometry is a math class that provides an introduction to all the branches of geometry. This branch of math is concerned with the study of points, lines, angles, curves, surfaces and solids. And when professors assign tasks, they can be tough sometimes.
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Geometry is a real challenge
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Out top 4 tips to solving the geometry proofs and tasks
There are just a few things that if you understand them and you can follow these tips, it’ll make the geometric proof a little bit easier.
All geometry proofs are challenging in that you’ve got to think. But there are some things that you can do to make life easier. So five tips to solving any geometry proof.
Tip 1: Don’t forget the properties
Hey, no, the postulates, theorems, and definitions. And you know what I need to add to that as well. Don’t forget the properties. The properties. Look, here’s the reality, you have got to know the postulates, you got to know the theorems, you’ve got to know the definitions and you’ve got to know the properties. You just have to. You just have to. These are things that you need to keep reviewing.
You need to keep having it in your mind. And here’s the reality, to do a proof. You need thoughts in your mind. Where do the thoughts in your mind come from? They come from the postulates, from the theorems, from the definitions, from the properties.
These give you the thoughts of the proof. If you don’t have them in your head, then you’re going to be in trouble. Now, when I taught geometry, I printed out a sheet for every student. It listed all the postulates, all the theorems. It listed the key definitions of the course. And it also listed in the section the properties.
This was a quick reference. She and the students could look at it and go through and refresh their memory on the postulates, theorems, definitions, and properties, because, again, these are the thoughts that you need to be able to write the proof. So, listen, I can’t state more emphatically or emphatically enough that in order for you to be able to write a geometric proof, you have to have the thoughts necessary and where those thoughts come from, the postulates, theorems, definitions, and properties.
So maybe you don’t have that sheet. You do have a book. The book probably lists all your postulates and theorems, hopefully in order. The book should also have an area where you can look up definitions and there should be somewhere a concise list of your properties. You need to refer to these often when doing proofs. It’s really the starting point. You got to know your postulates. You got to know your theorems. You got to know your definitions. You’ve got to know your properties. You’ll see that in a moment when we do our proof here.
Tip 2: Always label the drawing
It’s a big tip. It’s an important tip. So I’ve got a proof down here. It says, Given line segment essay is parallel line segment TB and always the midpoint of Abbey. OK, it’s not a ton of labeling on this opening proof, but there is some. This is what I like to do.
And extend the line essay you I like to go through. And extend line TB again, showing me the parallel lines, that’s important also then oh is the midpoint and I’m just going to put a dot there for oh, and give that little midpoint my midpoint symbol there.
OK, so but label drawing, when you have conquerer and angles marked the angles as congruent when you have congruent sides marked the sides as congruent. The picture is a visual to help you to think through the proof label. The drawing label. The drawing label. The drawing.
Tip 2: Know where you are going
Know where you’re going. You got to know where you’re going in any geometric proof. You just have to know where you’re going. You know, if I should get in a car and go to Oshkosh these days, you’d probably get there because you’d get in your car, you’d punch it into your GPS, and off you would go.
But in reality, in the old days, you had no earthly idea where Oshkosh was. You’d never get there. You’d never find it. If you didn’t know it was in Wisconsin, you’d never head there. Look, you got to know where you’re going. You got to know where you’re going. You’ve got to know where you’re going.
So you might say, well, where are we going? You’re going toward what you’re trying to prove right in this example here. Let me just focus it here if I get it not working.
So where are we heading in this proof? We’re trying to head toward a triangle. Toby is congruent to the triangle. So there we go.
That’s where we’re heading. We’re supposed to prove triangle. Toby is congruent, the triangle. So it’s important to know where you’re going. So let me tell you, this is my thought then because I know where I’m going. I always, in my proof, come down at the bottom. I usually give myself enough space and sometimes it’s further down than it needs to be. And that’s OK.
But I right where I’m going. Triangle top is congruency, triangle Essawi. Now, there’s no number there because I don’t know how many steps it’s going to take to get there. I don’t know. I don’t know how many steps it will take to get from the given to what I’m trying to prove.
But I do know that’s where I’m going. But here’s the key. All right, think about it. I have to prove to triangles to be congruent. Now, this point in our geometry, we would have learned four ways of true approving two triangles to be congruent. So, again, you got to know your postulates.
You got to know your theorems. So can you think of any ways of proving triangles to be congruent? I hope you can. I’ll give you the first one. How about S.A.S.?
Side, angle, side. Right, if the side of one triangle, if the two sides of one triangle, the included angle, are congruent to the corresponding two sides and included angle of a second triangle, then the two triangles are right side, angle, side.
Can you think of another one?
Were you thinking side, side, side? If three sides of one triangle going to three sides of another triangle, then the two triangles are congruent.
Side, side, side. How about angle, side, angle? If two angles of one triangle in the included side are congruent to the two angles included side of the second triangle, then the triangles are congruent. Hey, there’s one more, one more denote. How about the angle, angle, side, two angles of one triangle and a third and a side are congruent to two angles of another triangle and aside, then the two triangles are congruent.
Look, that’s where we’re heading. That’s where we’re going, we have to prove to Triangle’s congruent what means do we have to prove to the triangle? We’ve got citing a side, side, side, side, NGOs, side, angle and hang alongside. Those are the four possibilities. It’s going to be one of those four. That’s where we’re heading. So that’s an important tip. Tip number three, know where you are going? No, where you are going. You got to know where you’re going or you’ll never get there.
So let’s quickly recap number one. No, you’re PoshTots Serum’s definitions and properties. No Label drawing. Number three, know where you’re going. All right. Number four gets used all the time. The given is always given for a reason.
The given is always given for a reason. Let’s go back down to our proof. So, as you know, to start a proof. Statement No one is going to be your given information. All right, basically rewrite my segment to say is parallel line segment TB and oh is the midpoint of Abbey and of course, reason. Is the ever-popular given? OK, but here’s the point, giving is always given for a reason. I have two parts to the given, right? There are two things here. I have parallel and I have a midpoint. I have parallel and I have a midpoint.
And I can use those two things that the givens are always given for a reason. Now slide down a little bit here. Remember where we’re heading. We’re heading toward a side, angle, side, side, side, side, angle, side, angle or angle, angle, side.
We are trying to get angles congruent and sides congruent. That’s what we’re trying to do. So can I use parallel to do that? Well, again, tip number one, you’ve got to know your postulates, theorems, definitions and properties, so I look at my drawing, I see my blue parallel lines. Do you see my blue parallel lines? I see the blue parallel lines. You know what else I see? I see a transversal. You see that transversal.
And you know what else? Because I know my passions and theorems, I know that perhaps I can’t completely circle. I’m trying to get away, but I know that angle to me this way.
I know that angle too is congruent to angle five. Right. Because I know the theorem that the alternate interior angles. Of parallel lines are congruency. So I know that angle five is the congruent angle to. But you know what, that’s not the only thing. Look, here’s another transversal of these two parallel lines right there. You see that? So, therefore, then here’s another pa.
Of alternate interior angles, right? So not only can I say angle five is congruent to Angle, too, I can also say angle six is congruent to angle one. You follow. So angle six, you get rid of this again, is congruent to angle one. So again, what would be the reason? Again, you got to know your definition. You got to know your postulates. You got to know your theorems. Right.
And we know that given two parallel lines, we’ve got we’ve got to postulate that says the alternate interior angles are congruence and therefore that’s our reasoning alternates. Your angles are congruent. Again, if if your teacher wants you to give a little bit more then you’d give a little bit more. Right. But give in parallel lines, the alternates, your angles are congruent. OK, now let’s go back to the labeling aspect. So I’m going to label I have five congruency to write, five congruency to, and I’m going to come over to my proof, a pillow, a four angle. Who what else use a little color here. I have a single one congruent to Àngel six Àngel one congruent angle six.
And I have a second angle now. They both use the same reasoning. And so I went that way. Right. So now I have two angles. I’m looking, where am I heading? Right. You’re always thinking, where am I heading, where am I heading, where am I heading? Or how about angle. Side angle looks promising doesn’t it? If I can get a line segment as congruent line segment tbh that would be nice if I can get there. There’s no such thing as angle, angle, angle by the way you might say, well let me say angle three and angle four are congruent because you know you’re theorems and you know that vertical angles are congruent. And you know, your postulates, you know, vertical things are congruent. Well, there’s no such thing as angle, angle, angle, right. We’ve already taken care of that in the beginning by stating where we’re heading. There’s no angle, angle, angle. But there is an angle, side, angle, and there’s an angle, angle side. And who knows that maybe we still might use a side angle side, so we got a couple of possibilities going. Obviously, the more viable ones seem to be angle, side, angle and angle, angle, side. OK, so let’s go back to tip number four.
Tip 4: The given is always given for a reason.
The given is always given for a reason. And so let’s go back to our given. What else is in the given? Hey, we’re given that oh is the midpoint of AB.
Always the midpoint of Abby oh, wait a minute, think back to midpoint, that’s a definition. Do I know my definitions? I hope you know your definitions. Do you know what the definition of midpoint says? I hope you do, because here’s what it says. If OT is the midpoint of Abbey than a AOL equals Obbie, a O equals Obbie again. What’s the reason? This is simply a definition definition of midpoint. Now, at this point, some students want to end the proof.
And say that the two triangles are congruent and it’s not time to end the proof yet, I have already shown. One angle of one triangle can’t go to another, the second angle of one triangle congruent to the corresponding of another, but I have yet to prove a side congruent and again, to be a you know, if you’re going to be technical and some teachers don’t want you to be as technical, I always want my students to be technical. We really need to go from equal right.
Eddowes a distance, distance from Aido equal the distance from OTB and therefore we need to end up with congruent. But that’s an easy task, right? We can go and say line segment AOH is congruent to line segment Pobby and again, because you know your definitions I can’t stress enough. You’ve got to know the postulates, theorems, definitions and properties. How can you have the thoughts. Do approve if you don’t know the stuff to think you got to know the stuff to think.
That’s just the definition, it’s definition of congruent. And again, I’m going to come back here now and I’m going to label I now have a side. I’m going to label my drawing. I have this side congruent to this side. Hey, look what I have looked back at the drawing.
I’ve got. Angle, angle, I don’t I, I absolutely do, you know what that means, proof done. I am done now. I now have proven that triangle job is not congruently triangle s.o.s a by. Angle Angle sighed and the proofs completed now I will say this, some that are very, very technical would want you to switch around your order here and here on at least one of them, because if you look and sometimes even use symmetric properties, switch sides, because in your triangles, Obbie is on the left and always on the right. So one they’d want you to use Symmetric to get the Obbie on the left and the arrow on the right. And then they like you to use definition of a distance to switch from AoE to okay. Again, if your teacher wants you to be that technical, I think that’s great. If you’re going to end up being a lawyer one day, that’s very good for you because look, here’s the really good lawyer state. The simplistic good lawyers tell the jury things that they should already know and you would assume they know, but you’re going to tell them anyhow because you going to make sure they know.
And so I never have a problem with a teacher being very, very technical there. I think it’s a good thing. Hey, there we are. We just did a proof five stepper. And you know what we did? It basically was that given the give and got us all the way to the end because we realized the gibbons always given for a reason and we knew where we were heading and there we were again. I understand you got to know the PoshTots theorems, definitions and properties.