Mentor: Today we are going to go more in-depth with angles learning terms for kinds of angles and how to find the degrees of angles in certain situations.

Student: So we will be doing more things like Acute and Obtuse angles?

Mentor: Right, the first one we will be starting with are supplementary angles. Do you have any guesses about what supplementary angles are?

Student: Well, are they angles that are more than 180 degrees?

Mentor: No, but that's a very good guess. Supplementary angles are angles that add up to 180 degrees, so that if you put them all together they would form a straight line at the bottom.

Student: So two right angles would be supplementary?

Mentor: Good, can you think of another example?

Student: Um...100 degree angle and an 80 degree angle.

Mentor: Good, it looks like you've got the hang of that one. The next one that were going to learn are complementary angles. Do you have an idea of what those might be?

Student: Angles that add up to 360 degrees?

Mentor: Almost, they are angles that add up to 90 degrees. Why don't you come up with a couple sets of complementary angles.

Student: Well, 45 and 45, 30 and 60, 30 and 30 and 30....

Mentor: Good, looks like you have that one down too. Now we are going to move on to something different called vertical angles. Do you have a guess what vertical angles are?

Student: Well, are they angles are that are straight up and down?

Mentor: Good guess, you thought of what vertical means and applied it to angles. What vertical angles really are is sort of like that. Vertical angles are angles that are formed by the same two lines so one angle is one side of the intersection and the other angle is the other side. In the image below angles a and d are vertical and angles b and c are vertical. Can you guess what is the relationship between two angles if we know they are vertical?

Student: They're equal!

Mentor: Good, but when we are dealing with angles we call equal congruent. Can you tell me why they are congruent?

Student: Well, they look like they're equal.

Mentor: That's true, but unless your trying to estimate you need proof before you believe something in math. Can you think of a proof of why vertical angles are congruent?

Student: Well, it would be very counter intuitive if they weren't.

Mentor: True, but let's try and think of something more concrete. Let me put you along the right track. What can you tell me about angles a and b in the image above?

Student: Um...They're Supplementary angles!

Mentor: Good! Now what can you tell me about angles b and d?

Student: They're supplementary too!

Mentor: So tell me what we know about angles a, b, and d.

Student: Well, a+b=180 and b+d=180.

Mentor: If we know that a+b and b+d both equal the same thing what can we say about them?

Student: I guess a+b=b+d.

Mentor: Exactly right! Now, are you familiar with the transitive property?

Student: No, what is it?

Mentor: The transitive property states that if a+b=b+c then a=c. Do you understand why that works?

Student: Yes! That's like if you have 10 apples, and you take away two from b oth sides, it's still 8 for both sides.

Mentor: Right, if you have x number of apples for each side and you take away y, both sides are x-y. You can also solve it using very simple algebra, a+b=b+c, subtract b from both sides and you have a=c. So knowing the transitive property what can you tell me about the vertical angles we were working with?

Student: Well, we knew a+b=b+d so a must equal d!

Mentor: Great! So you can see that knowing only the one simple rule about supplementary angles and figuring out the transitive property we were able to discover another rule. Let's see how much we can figure out just based on one rule and logic.

Student: Is all geometry like this?

Mentor: This isn't only this type of problem, but almost all geometry is heavily interconnected all stemming from a few simple rules.

Student: Cool!

Mentor: From now on today we will be working with the diagram below. The first thing that we must know about the diagram is that lines q and r are parallel. Can you tell me what parallel lines are?

Student: Aren't parallel lines lines that don't intersect?

Mentor: Not quite, there are also lines called skew lines that don't intersect, a skew line is when one line is going left to right in one plane and another is going up and down in a different plane. Can you revise your answer to exclude skew lines?

Student: Lines in the same plane that don't intersect?

Mentor: Good, another important thing to know about parallel lines is that they have the same slope. The line that goes through the parallel lines, line s is called the transversal. The transversal is a line that intersects with both! paralle l lines. So lets review what we already know, what are angles e and f in relation to each other?

Student: Supplementary angles.

Mentor: What does supplementary angles mean?

Student: That they add up to 180.

Mentor: Very good, so what are angles f and g?

Student: They're vertical angles which means they are congruent.

Mentor: Perfect. Now were going to learn about corresponding angles. Corresponding angles are angles that are in the same place relative to the nearest parallel line. Can you give me an example of corresponding angles?

Student: Angles a and e?

Mentor: Good, angles c and g, b and f, and d and h are all also sets of corresponding angles. Can you guess what corresponding angles are in relation to each other?

Student: They're congruent!

Mentor: Good, can you tell me why they're congruent?

Student: I don't know, they just look like they're congruent.

Mentor: Remember in math that's not good enough, you have to prove it. Does it make sense to you that if you have two sets of lines with the same slope for a line in one set, and a line in the other set, then it would come out with the same angles for each set?

Student: I think so.

Mentor: Try imagining it in your mind, take one fixed line, and slide the other up and down it, you always come out with the same angles right?

Student: Yeah!

Mentor: So if we know that, can you now come up with a proof for why corresponding angles are congruent?

Student: Yes! Since q and r are parallel they have the same slope, and s is just one line, so it has the same slope, so the angles will always be congruent between the two parallel lines.

Mentor: Very good, so now we know that corresponding angles are congruent. Now! let's m ove on to Same side interior angles and alternate interior angles. Can you give me a set of Same side interior angles?

Student: How about d and f?

Mentor: Good, now can you tell me what the relation between these angles are?

Student: Are these congruent too?

Mentor: No, let's find out why they are not congruent, and what they actually are. So, remember how I said we would use the same basic rules? Were going to do it again.

Student: OK, so how do we start?

Mentor: Well, what is the vertical angle of angle d?

Student: Angle a.

Mentor: Good, so what's angle d in terms of angle a?

Student: d=a.

Mentor: Good, now what's angle a in proportion to angle e?

Student: They are corresponding angles so a,=e.

Mentor: Good, so now we know d=a and e=a, what knew information can we take from that?

Student: d=e!

Mentor: Right, because remember if they both equal the same thing, they must be the same thing.

Student: We also know that e is supplementary to f!

Mentor: Very good, now take it all the way back again and tell me what d and f are to each other.

Student: They would be supplementary right?

Mentor: Good, so if d was 110 what would f be?

Student: Um....70!

Mentor: Good, now can you tell me another set of Same Side Interior Angles?

Student: c and e?

Mentor: Good, now lets move on to Alternate Interior Angles, can you give me a set of those?

Student: How about g and d?

Mentor: No, because remember Alternate Interior Angles, g is on the exterior.

Student: So would it be d and e?

Mentor: Good, so let's see if you can figure out the first step of discovering the relationship between d and e.

Student: Well, would it be a is the vertical angle of d so a=d?

Mentor: Good, what's the next step?

Student: Well, a is the corresponding angle of e, so a=e.

Mentor: Very good, now piece the information together.

Student: Well, if a=d, and a=e, then d=e.

Mentor: Very good, see you don't even need me, using logic and the one basic rule of supplementary angles you can figure it out for yourself! Now you never need to worry about forgetting these rules because you can always figure them out again for yourself.

Student: Cool! If I ever forget something I can just work back to figure it out!

Mentor: That's the best part of geometry. Now let's practice a litt! le, tell we what they are in relation to each other, and why. so lets start with b and g.

Student: But those aren't any of the rules we've gone over!

Mentor: That's true, but remember, it's all just basic the basic rules, you just haven't learned the name of it yet, you still know how to do it.

Student: Well, OK, so b=c because of vertical angles and c=g because of corresponding angles, so b=g!

Mentor: See, if you just think it through and figure it out instead of relying on memorized rules you can find out anything! Now let's try h and c.

Student: OK! So h=e, because of vertical angles and then c is supplementary to e because of same side interior angles. So, h is supplementary to c.

Mentor: Very good, so always remember that you shouldn't despair because you don't know a rule, you can just figure it out for yourself. A good way to practice doing this is by going to the Angles activity.

Student: Wow! I never knew geometry could be so fun!