Finding triangle angles with algebra

0:00- [Instructor] So we're told in triangle QRS, these are the angle measures. So they give us the measure of QRS. So QRS, so that's that angle measure right over there. I'll color code it. So they tell us it's Y degrees, so I'll just write Y degrees right over there.

0:18And then they tell us the measure of angle SQR. SQR, so that's this angle right over here. It is 3Y-18 degrees. This one right over there. Find the measure of angle SQR. So this purple one right over here. Pause this video and see if you can have a go at that before we do this together.

0:47All right, now let's do this together. So the way my brain approaches this is okay, they've given us at least an algebraic expressions for two of the angles of a triangle. And then they also tell us that this is a right triangle so that third angle is 90 degrees.

1:02So I can think about the three interior angles of a triangle that add up to 180 degrees and maybe I could use that to set up an equation to then solve for Y. And then if I know Y, I can figure out what this angle is going to be. So let's do that. So we have angle SQR, which is 3Y-18 degrees.

1:24We'll just assume everything is in degrees right over here. And then we have plus the measure of angle QRS, which is right over here. So +Y plus this right angle plus 90 is going to be equal to 180. Everything's in degrees here. And so let's just solve this algebraically.

1:46And if you didn't do this the first time, this is also another good time to pause this video and see if you can solve for Y. Alright, so let's just combine some like terms. We have three Y plus Y is going to be four Y and then we have minus 18 or we're subtracting 18 plus 90.

2:0690 minus 18 is 72, so plus 72 is equal to 180. Now we could subtract 72 from both sides. So I just have my Y's on one side and then I am going to have, these are going to cancel out and I'm going to have four Y is equal to 180 minus 72 is 1 0 8. And I can then divide both sides by four to figure out what Y is.

2:42And so Y is going to be equal to, let's see, a hundred divided by four is 25. And then I'm gonna have two more fours in there. So this is 27, y equals 27. Now you might be tempted to say you're done, but all we figured out is the measure of angle QRS. We figured out this one.

3:01This one right over here is Y degrees or 27 degrees, but the angle we're trying to figure out is three Y minus 18. So let's do that. Let's figure out three Y. We know Y is 27 minus 18. That is going to be equal to, let's see, three times 27 is 60 plus 21. Let's see, that's 81.

3:24This first part right over here and then we're going to subtract 18 from that. Did I do that right? Yeah. Three times 20 is 60, 3 times seven is 21, you add 'em together, 81. Alright, and then 81 minus 18 is going to be equal to, let's see, 81 minus 20 is 61 and this is gonna be two higher than that.

3:44So this is going to be 63. So the measure of angle SQR is equal to 63 degrees. Let's do another example. So here they're telling us in isosceles triangle, JKL, these are the angle measures measure of KLJ, KLJ is, so that angle right over there. I need a different color, is 2B plus seven degrees.

4:19And then they tell us measure of LJK, LJK, this angle right over here, that measure is 6B plus six degrees and they want us to figure out the measure of angle J, K, L. So they're trying to figure out what this one over here is. Lemme do that in a different color.

4:43So let's put this as a question mark. So how can we do this? Well, we might say, okay, well if we had an expression for this angle right over here, we could then do the same thing we just did in the last example. Add those three expressions together, the interior angles, the measures of them need to add up to 180 degrees.

5:04But what is the expression for this? And the clue here is this is an isosceles triangle. They mark it here. The length of this side is equivalent to the length of that side and you might already know. And if you don't already know, you're about to know that if you have an isosceles triangle, the base angles have equal measure.

5:22So if the measure of this one is 2b plus seven degrees and the measure of this one is also going to be 2b plus seven degrees, and now we can do the same thing we did in the last example, we can add these three expressions up and say they need to be at, they need to equal 180 degrees.

5:41So let's do that. We have 2b plus seven plus another 2B plus seven plus 2B plus seven plus 6B plus six plus 6B plus six is going to be equal to 180 degrees. Now let's add up all of the B terms and a color I haven't used yet. Two B plus two B2B, to be or not to be, 2B plus 2B plus 6B, that's going to be 10B.

6:14And then I am going to have seven plus seven plus six. That's 14 plus six is 20. That's a nice number that we should be able to do the math with pretty easily is going to be equal to 180. And now I can subtract 20 from both sides. Two, get these cancel out, I get 10B is going to be, I'll switch back to black, is going to be equal to 160.

6:47Divide both sides by B, sorry, divide both sides by 10. We want to isolate the B, divide both sides by 10. And we're going to get B is equal to 16. And so to actually find this, this angle measure, we just have to evaluate 2B plus seven. So two times 16 plus seven.

7:08This is 32 plus seven. This is 39, so this is 39. The measure of this angle is 39 degrees, which by the way is the same as this one right over there. And we're done.