Hi everyone! Today we're going to be talking about angles. Our objectives for today are that students will be able to measure and classify angles, you'll be able to identify and use congruent angles and the bisector of an angle, and you'll be able to use the angle addition postulate. Let's get started!
Here's a real world example of when we use angles in real life, alright? So one of the skills that Dale must learn in... Carpentry class is how to cut a miter joint. This joint is created when two boards are cut at an angle to each other, and it looks something like this. Alright, we've got an angle and an angle, and then it forms a right angle.
He has learned that one little miscalculation in an angle measure can result in mitered edges that do not fit together. So this is one example of why angle measurements are so important. Alright, let's go through some key vocabulary when talking about angles. So an angle is formed by two rays with a common end point. So for example, in this angle right here, we've got our common end point would be B, and we've got this ray right here, ray AB, and ray BC.
The common end point is called the vertex. Alright, so in this case, B would be the vertex. The rays are called... the sides of the angle and we always name an angle using three letters. So the middle letter must always be the vertex.
If you use a single letter, you can use a single letter if there is only one angle located at the vertex. So, and when referring to the measure of an angle, we always use the lower case m. That means measure. So for example here, The measure of angle ABC and B has to be the middle letter because B is the vertex equals 60 degrees.
Alright, now we have four different types of angles and we're going to go through all four of them together. So I just kind of want to zoom in so it's easier for me to write here. So the first angle is called an acute angle. And an acute angle is when our angle is less than...
90 degrees. Alright, it's a cute little angle. Then an obtuse angle is our second kind of angle and that's when the measure of the angle is greater than 90 degrees. Alright, think of the word obese meaning means big, right?
It means large. Obese, obtuse, they're kind of similar. So I always think of an obtuse angle is the bigger angle, alright? Like obese. And then our third angle is a right angle.
And a right angle always equals 90 degrees exactly. And this little square right here will always indicate that it's a right angle. Last but not least, we've got a straight angle. And a straight angle is always equal to 180 degrees. And it's basically just a straight line.
The angle goes from here to here like a protractor and it's 180 degrees. Alright, I had to throw in a little joke slash cartoon in here for you guys. Alright, you can laugh or you can think it's stupid, but I thought it was funny.
So why is the obtuse triangle always upset? Because it's never right! And then I mean, come on, I had to put this meme in there too.
Alright, hopefully it lightens up the video notes for today a little bit. If you don't get it, ask a friend, but I thought it was funny. Alright, back to our examples, back to our notes. So example one, name the vertex of the angle.
The vertex of this angle would be K, and we just use a single capital letter. Name the sides of the angle. The sides of the angle would be the rays. So it would be ray KJ, and we put a little line with one arrow above it, and it would be ray KL, and we indicate it with a little ray above it.
Three different ways we can name this angle. We can name it angle K, just like that, and this little symbol right here means angle. So we call this angle K because it's the angle. only, we can use a single letter because there's only one angle located at vertex K.
Alright, there's no dispute about which one it is. We can also name it angle. j, k, l. So k has to be in the middle because it's the vertex. Or you can name it angle l, k, j.
All three of those would work. And if we wanted to classify this angle, we classify it as an obtuse angle because it's greater than 90 degrees. All right, and greater than 90 degrees meaning it's bigger than a right angle. Example 2, if we wanted to name the vertex of this angle, it would be S. S is the vertex.
Name the sides of the angle. It would be ray SR and ray ST. Three different ways to name the angle. We can name it angle S, angle RST, and angle TSR, as long as the vertex is the middle letter.
It doesn't matter which order you do it. all three of these would be correct ways to name the angle. And then to classify the angle, this is a right angle. And we know because of this little square right here, that indicates that it's exactly 90 degrees and it's a right angle. All right, congruent angles.
So basically what we can look at when we have two angles is if the measures are equal, then the angles are congruent. And the way we would write this is we would say if the measure of angle A is equal to the measure of angle B, then the angles are congruent. And we would write this as angle A is congruent to angle B. And just a refresher, the congruent symbol is an equal sign with a squiggly line on top.
That means congruent. In our example here, angle A is 75 degrees, angle B is 75 degrees. Their measures are equal, so therefore the angles are congruent. Alright, lots of new vocabulary today.
So an angle bisector. An angle bisector is when you have a ray that divides the angle into two congruent angles. Alright, so an angle bisector is when you have a ray that divides an angle into two congruent angles.
So in the diagram on the right, ray BD, alright, that's this ray right here, is an angle bisector. Therefore, we know that angle ABD is congruent to DBC. And we would write this as...
Angle ABD is congruent to angle DBC. Okay? So that's when an angle bisector just cuts an angle in half.
Alright, perpendicular lines. Perpendicular lines are when we have two lines that intersect at a right angle. Alright, so perpendicular lines are when we have two lines that intersect at a right angle.
The symbol for perpendicular is basically an upside down T. It looks like this. Alright, that means perpendicular. If you want to see it bigger, perpendicular.
Alright, and the diagram on the right, we would say that line L is perpendicular to line M. And that's how we would write it. We know that they're perpendicular because this little square right here indicates that they meet at a right angle. It means they form a right angle. That's how we know they're perpendicular.
Now a perpendicular bisector. A perpendicular bisector is a line, segment, or ray that's perpendicular to a segment at its midpoint. Alright, so that means it's perpendicular and forms a right angle, and it cuts the segment in half because it's perpendicular at its midpoint. So in the diagram on the right, Lm, line segment Lm, is the perpendicular bisector to line segment PQ. So they're perpendicular, meaning they form a right angle, and it cuts PQ in half.
Alright, let's go through example 3. So A, we want to write another name for CBF. Now CBF is right here, alright? We cannot just call it angle B, because we could be talking about this angle right here if we said angle B.
We don't know what angle B we're talking about, so we have to use all three letters to name this angle. Now we already used CBF, so we can call it FBC. That works, because...
b is the vertex and it's in the middle, you just switch the f and c around. Name the sides of EBD. So EBD, let's find it. EBD, right there, that's angle EBD.
So the sides of an angle are always rays, so the sides would be ray EB and ray BD. Now we want to classify angle. ABC so angle ABC is this right here and it's a straight line so therefore it is a straight angle give an example of an obtuse angle so you've got lots of different options here I'm just gonna go with let's do a B F that's an obtuse angle it's greater than 90 degrees alright so angle a BF is an obtuse angle there's lots of other Angles you could have said. You could have said angle EBD is also an obtuse angle.
Alright? Lots of options here. You could have said, let's see, EBC.
That would have been an obtuse angle as well. Alright, let's see. Name two congruent angles. So two congruent angles. I'm going to, first before we do this, I'm going to erase this so we can see this a little better again.
Alright. Now we want to name two congruent angles. We've got a couple options here. We know that this little indicator right here means it's congruent, alright?
And we know they're vertical angles, they're across from each other. Don't think we've learned vertical angles yet, but this is a quick preview. When angles are across from each other, they're congruent. These little lines right here also mean that they're congruent, so that's an indicator right off the bat. So we know angle EBA is congruent to angle CBF.
So let's say angle EBA is congruent to angle CBF. Now you could have also said, we know that this angle right here, angle DBC, is a right angle. That means that angle ABD also has to be a right angle because it forms a straight line.
They both have to be 90 degrees. So you could have said angle ABD is congruent to angle CBD. that would have worked as well.
Alright, and then last but not least, we want to name a perpendicular bisector. So a perpendicular bisector, we know that this is our right angle right here. So our perpendicular bisector would be BD because it cuts AC in half. We know that these little lines indicate that it's cut in half.
So it's D. So B is the midpoint and BD is the perpendicular bisector. Alright, any questions on that, please raise your hand, call me over, or ask a friend.
Otherwise, let's move on. Alright, example number four. I want you to try to do example four on your own.
So use the diagram. Do A through J. When you come back, I'll have the answers written on your screen and we will quickly go through it together. So go ahead and pause your video now. Alright, so here's the answers to example 4. Let's quickly go through them.
So name the vertex of angle 2. So we look for the 2, it means this angle right here. So the vertex is T. Name the sides of angle 4. So we look for the 4, this would be angle 4. The sides are going to be raised, so TW and TZ. Another name for angle 3. So angle 3 is this angle right here.
So another name for it would be angle UTZ or angle ZTU. Both of those would work. We cannot just say angle T because angle T could be talking about this angle here, could be talking about this angle here.
Because this vertex shares other angles, you cannot just say angle T. Alright, write another name for angle 1. Angle 1 is this angle right here. Would be... Y, T, X, or X, T, Y. Classify YTW. So we look for YTW.
It would be this angle right here. YTW is a right angle because of this little square right here that indicates that it's 90 degrees and it's a right angle. Classify YTU.
YTU would be this angle right here. We go from the Y to the T to the U. That's how we can identify the angle. And that is greater than 90 degrees so it is obtuse.
G, we want to classify X, T, U. X, T, U is this little cute acute angle right here. It's a tiny little cute angle. It's an acute angle because it's less than 90 degrees. And then W, T, X. So W to T to X, it's the straight angle right here.
It's 180 degrees. Because it's a straight line means it's a straight angle. Name two perpendicular lines. So we want them to form.
a 90 degree angle or a right angle. So it would be line YZ and WX. These two lines form a right angle.
And then if we want to name an angle bisector Ray TU is the angle bisector because these little lines indicate that angle 2 and 3 are congruent to each other therefore TU cuts it in half and is the angle bisector. If you have any questions on these call me over so I can help you out otherwise let's move on. Alright, the angle addition postulate. Basically what the angle addition postulate says is that if we have a ray that cuts it in half, alright? If D is in the interior, it's on the inside of angle ABC, and this cuts it in half, well not necessarily cuts it in half, it just kind of divides it, alright?
We can say that this angle plus this angle equals the whole angle. So the way we would write that is we would say, If D is in the interior of angle ABC, then angle ABD plus angle DBC equals angle ABC. So if this is on the inside and it divides the angle, we don't know if it cuts it in half directly, we don't know if it's an angle bisector, but we just know it's in the middle of angle ABC. Then this angle.
plus this angle equals the whole angle. It's kind of common sense, but there's a postulate that describes it, and it'll help us out in the long run. Alright, here's a few examples of when we're gonna need the angle addition postulate.
So if ABD is 48 degrees, so that's this angle right here is 48 degrees, and angle DBC is 78 degrees, we wanna find the measure of ABC. So all we need to do is add them together. 48 plus 78. Angle addition postulate. We're adding the angles together.
So ABC is 126 degrees. Now, if the measure of DBC, so we're on to number 2. I'm going to erase this because we've got a different problem now. If DBC is 74, so if this is 74 degrees, and ABC or the whole... thing is 119 degrees we want to find ABD so we can take the whole thing subtract what we know and that's going to give us this angle right here so we would do 119 minus 74 and that's going to give us the measure of ABD and that's going to be 45 degrees Alright, now we're going to throw some algebra into the mix. Alright, so with number 3, we're given that the measure of PQR, so that's the whole thing, is 141. And we want to find each measure.
So first we're going to find x, then we want to find the measure of each individual angle. So, first thing we can do is we know that this angle plus this angle is going to equal 141 because of the angle addition postulate. So, PQS is...
13x plus 4 plus SQR is 10x minus 1 equals PQR, the whole thing, 141. Now we can solve for x. So we're going to combine like terms. We get 23x, 4 minus 1 gives us 3, plus 3 equals 141. Subtract 3, 23x.
equals 138 divided by 23 Plug that into our calculator and we get that x equals 6. So we solved for x first. Now we want to go back and find pqs. So we plug x in to 13x plus 4. So 13 times 6 plus 4. And that's going to give us pqs.
So we plug that in. 13 times 6 gives us 78 plus 4. 78 plus 4 is 82. So PQS is 82 degrees. And then last but not least, SQR is 10x minus 1. So we plug x in to 10x minus 1. So 10 times 6 minus 1, 60 minus 1 equals 59. SQR is 59 degrees.
And you're done! Alright, number 5. If the measure of angle JKM is 43, so let's label what we know. So this is 43 degrees, and MKL is 8x-20. So MKL is this angle right here, and it's 8x-20 degrees.
And we know that the measure of JKL is... 10x minus 11. So the whole thing is equal to 10x minus 11 degrees. We want to find each measure.
So the first thing we want to do is write an equation to represent this. Angle addition postulate, we know that this angle plus this angle equals the whole angle. So an equation to represent this would be 43 plus 8x minus 20 equals... 10x minus 11. Now we're just solving. So combine like terms.
43 minus 20 gives us 23 plus 8x equals 10x minus 11. Subtract 8x from both sides because we want to get all the x's to one side. Add 11 to both sides because we want to get all the constants or our numbers to the other side. So 34 equals 2x.
Divide by 2. x equals 2x. 17. So we first solved for x. Now that we know what x is, we can go through and find mkl and the measure of jkl.
So mkl, we know is this angle right here, and it's equal to 8x-20. So we plug 17 in. So 8 times 17 minus 20. That's going to give us mkl. When we do that, we get...
Plug that into our calculator, that equals 116. Alright, so the measure of mkl is 116 degrees. And then jkl is the whole thing, and we know that equals 10x minus 11, so 10 times 17 minus 11, 170 minus 11, that equals 159 degrees. And that's it.
the measure of JKL. You've solved for all three of your values and you're done. Alright, I want you to try to do number seven on your own. So, read through the problem, label your diagram, write an equation to represent what you're looking for, and then find all of the missing values. So go ahead and pause your video now.
Alright. So you should have gotten that x is 7, the measure of angle TUW is 38, the measure of angle WUV is 65 degrees, and the measure of angle TUV is 103 degrees. So the first thing you want to do is read through the problem and label your angles.
Alright, so TUW is 5x plus 3, WUV is 10x minus 5. and TuV, the whole thing, is 17x minus 16. So angle addition postulate, this angle plus this angle equals the whole angle. We write an equation to represent that. Then it's just algebra, and we're solving for x. So we combine like terms, move all the x's to one side, all the constants to the other, and solve for x, x equals 7. Then to find the measure of these three angles, we go back and plug x in. So to find tuw we plug it into 5x plus 3 and get 38. To find wuv plug it back into 10x minus 5 and get 65 degrees, and then to find the whole thing, we plug it back into 17x minus 16 and get 103. Alright, if you have any questions, please raise your hand, call me over so I can help you.
Otherwise, let's move on to our last example of the day. Alright, number 9. So if the measure of angle ABF is 6x plus 26, so let's look for that angle. ABF is this angle right here, alright, and that equals 6x minus, or 6x plus 26. Then the measure of angle EBF is this little angle right here, alright, and it's not going to fit, so I'm just going to circle it, and we know that this one right here is going to be 2x minus 9. And the measure of angle ABE, that's the whole angle. whole angle is equal to 11x minus 31. We want to find the measure of angle ABF. So we're looking for this angle right here, but we first need to solve for x.
Then we can go back and plug it in to find the measure of ABF. So we want to write an equation. We know angle addition postulate, this angle plus this angle equals the whole angle.
So we can say that 6x plus 26 plus 2x minus 9 equals 11x minus 31. Alright, then combine like terms. 6x plus 2x, we get 8x. 26 minus 9 gives us 17. So plus, plus. 17. Alright, and then we know it equals 11x minus 31. Now we just want to move all the x's to one side, all the constants to the other. So subtract 8x, these will cancel out, add 31, these will cancel out over here.
So we've got 48 equals 3x, divide by 3. 48 divided by 3 is 16, so x equals 16. Now we're not done. The problem is not asking us to find the value of x. It's asking us to find the value of abf.
So we need to go back and plug it in. So the measure, oops, and you should put an m in front. All right, the measure of angle abf equals 6x plus 26. Now we want to substitute.
16 in for x, so 6 times 16 plus 26. 6 times 16 plus 26, plug that into our calculator, and we get 122 degrees. And that is our final answer. Alright, so I hope you learned a thing or two about angles today.
If you have any questions, please call me over so I can help you, or turn to a friend and ask. Otherwise, look to the board to see what to do next, and I hope that you have a great day.