Welcome to a lesson on pairs of angles. More specifically we'll be talking about
complementary, supplementary and vertical angles.
As well as solving problems involving pairs of angles.
Let's first talk about adjacent angles. Adjacent angles
share the same vertex and share one side but do not
overlapped. So looking at this diagram here,
angle 1 and angle 2 are adjacent angles.
Well remember we can identify angle 1 as angle ABC.
And we can identify angle 2 as angle CBD.
And these 2 angles are adjacent angles.
I also wana point out there's a third angle.
sketched here. It would be angle ABD.
and the measure of angle ABD would be the sum
of the measure of angle 1 and angle 2. Looking at the second diagram here
angle 3 and angle 4 are adjacent angles. Which could also be identified by angle EFG
and angle HFG. Remember when identifying an angle using this notation,
is important that the vertex be in the middle.
Complementary angles are two angles that add to 90 degrees.
Looking at that angle here formed by the two black rays
Notice this little square here indicates it's a right angle.
Which means it measures 90 degrees. And angle 5 and angle 6
are two adjacent angles that form the right angle.
And therefore, angle 5 and angle 6 are complementary.
supplementary angles add to 180 degrees.
So if the angles where adjacent, as we see here,
they would form a straight line. And therefore are also called, a linear pair.
Angle 7 and angle 8 are supplementary angles.
The last special pair of angles we'll take a look at are vertical angles.
Vertical angles are two non-adjacent angles
found by intersecting lines. So looking at this diagram here,
notice that angle 1 and angle 2 are adjacent angles.
But number 1 and number 3 are non adjacent angles
formed by 2 intersecting lines. So angle 1 and angle 3 are vertical angles.
And angle 2 and angle 4 are vertical angle.
And all vertical angles are congruent.
So angle 1 is congruent to angles 3. Which means they have the same measure.
And there's a couple ways of indicating 2 angles that have the same measure.
1 way is by the number arcs. So if I use 1 arcs for angle 1,
and 1 art angle 3, we know they have the same measure.
And angle 2 an angle 4 are congruent. So again, using the arc method I could
use 2 arcs
for angle 2, and then 2 arcs for angle 4.
showing that those 2 angles are congruent. The other way to show that
2 angles are equal in measure would be to use hash marks or tick marks.
So if I wanna show that angle 1 and angle 3 are congruent,
I would put 1 arc for both, and then put 1 hash mark
through this arc. And 1 hash mark through this arc.
Showing those 2 angles have the same measure. And then for angle 2 and angle 4,
I would also use one arc.
But I have to use 2 hash marks here, and 2 hash marks here,
to show that those two angles are equal in measure. Now let's go ahead and take a look at some
problems,
to reinforce these special types of angles.
Here we have two intersecting lines and therefore two pairs of
vertical angles. And where asked to determine the measure of each angle.
So if we know these 2 angles are vertical, they must be equal in measure.
And we can use this information to determine the value of X.
Since there vertical angles, X plus 16
degrees must equal 4X minus 5 degrees.
So we'll go ahead and solve this for X.
subtract X on both sides. So we have 3x
minus 5 equals 16. Now I'll go ahead and add 5 to both sides.
Well if we had 5 to 16, well have 21.
Now if we divide both sides by 3, we know that X
must equal 7. So if X is equal to 7,
this angle here, would be
7 plus 16 degrees. Or 23 degrees.
And therefore because these angles are vertical angles, this angle here must
also be 23 degrees.
Which you could verify by subing in X equals 7 here if we wanted to.
Now the remaining 2 angles are also vertical angles.
So this angle here would equal this angle here. To help us determine the measure
of these last two angles,
notice that the angles along this line here would be
linear angles and therefore supplementary. So if this is 23 degrees,
and this angle here must be 180 degrees
minus 23 degrees. So that's going to give us 157 degrees.
If this angle is 157 degrees and then, so is this one here.
let's take a look at this last problem based upon this diagram here.
The first question is, name one pair of vertical angles.
So we're looking for two angles that are formed by
two intersecting lines that are not adjacent angles.
So this angle here would be a vertical angle with this angle here.
So we can say that angle INJ,
and angle MNL,
are vertical angles. The next question asks,
name 1, linear pair of angles.
So if we take a look at this line here,
the two angles that form this line would be angle INM. And angle MNL.
Notice that the linear pair would also be supplementary.
Next, we want to name two complementary angles.
Remember complementary angles have a sum of 90 degrees.
And since these 2 angles here form a 90 angle,
angle INJ and angle JNK,
would be complementary.
next question, name 2 supplementary angles.
And since we already mentioned a linear pair of these 2 angles here, are supplementary,
let's see if we can find 2 more supplementary angles.
Again, referencing this line here, we could say that angle INK,
an angle KNL are also supplementary,
because together they form a straight angle.
And for this last question, it says given that angle INJ is 61 degrees,
find the measure of the following 4 angles.
So if angle INJ is 61 degrees,
that would be here, let's see if we can find the measure of angle JNL. Well angle JNL,
and the given angle form a straight angle, and therefore there supplementary.
So angle JNL would be 180 degrees
minus 61 degrees.
So it would be 119 degrees. Were asked to determine the measure of angle KNL, which is here.
Well notice that angle KNL and angle KNI,
are supplementary and form a straight angle. And angle KNI is 90 degrees .
Therefore, angle KNL must also be 90 degrees.
And then angle MNL.
MNL is here, well it's a vertical angle
with the given angle, therefore, angle MNL
must be 61 degrees.
Now the last angle is MNI.
which is this angle here.
Well angle MNI and angle LNM
are supplementary. And since we have just said this is 61 degress,
angle MNI must be 119 degrees.
And the last thing they mentioned here is notice that MNI
and angle JNL are vertical angles and therefore there
equal. And we found the measure angle
JNL in part A. I think we'll stop here for this video.
I hope you found this helpful. Thank you for watching.