Today, we are learning about the triangle
angle sum theorem.
We know that the sum of all the measures in
a triangle have to add up to be 180.
So the definition of our theorem is that the
sum of the measures of the angles of a triangle
is 180.
When we look at the actual triangle, we have
angles A, B, and C. Angle A plus angle B plus
angle C will always equal 180 degrees.
That's look at an example problem.
We need to figure out all of the angles for
each of the numbers listed.
Angle 1 has to be 90 degrees.
Angles 2 and 3 also have to be 90 degrees
because they are supplementary angles, meaning
they add to be 180 degrees with angle 1.
We can figure out angle 4 which makes a vertical
angle with 130 degrees, vertical angles are
congruent so angle 4 is 130 degrees.
Another vertical angle we have is 8 and 40
degrees.
so that means angle 8 is 40 degrees as well.
Inside we have a triangle, with angles which
are 90 degrees, 40 degrees, and angle 6 which
is missing.
Based on this theorem, we know that three
angles have to add to be 180 degrees.
So that means 90 plus 40 plus x = 180.
90 plus 40 is 130, so when we subtract 130
to both sides, we find that x is going to
equal 50 degrees.
Last ones we need to figure out are 9 and
7.
These two are vertical so they will be equal
to each other, but since 40 and angle 9 are
supplementary so they need to add to be 180.
Angle 9 will be 140 degrees and angle 7 is
also 140 degrees.
Lets look at the different ways we can label
and classify a triangle.
We can label them based on their angles and
their sides.
Based on angles we can have equiangular which
means all angles are congruent.
An acute triangle means all angles are less
than 90 degrees.
We can have an obtuse triangle were only one
angle is obtuse, meaning it's greater than
90 but it is also less than 180 degrees.
The last one is a right triangle and that
is where one angle is a right angle which
means it is 90 degrees.
Classifying by the sides, we have equilateral
means all sides are congruent.
Isosceles means we have 2 sides that are congruent.
And our last one is scalene which means no
sides are congruent.
Let's look at three different types of triangles
to determine what they are.
If we know we have a triangle with sides 8,
4, and 5 that means we are labeling it based
on our sides and none on them are the same,
so that means it is a scalene triangle.
The next one, we are doing the sides again
so we are looking at these groups, we have
5, 5, and 8 so two sides are the same, so
that means we are working with an isosceles
triangle.
The last one, we are looking at angles.
We have 100 degrees, 30 degrees, adn 50 degrees.
Since one is obtuse, we have an obtuse triangle.
Let's draw these.
If we have an acute scalene triangle, scalene
means no sides have the same length, so these
are all different lengths and the angles are
all less than 90 degrees.
An isosceles right triangle, a right triangle
means we know we have a 90 degree angle.
Isosceles says we have two sides that are
congruent to each other.
So this is an isosceles right triangle.
We have an angle greater than 90 less than
180 and each side is the same length.
All three sides are congruent to each other.
And this is our obtuse angle.
We are still sticking with triangles.
Exterior angles are formed by a side and it
is also an extension of an adjacent side.
So if we look at an example of a picture,
angle 1, 2, and 3 in our triangle.
Angle 1 is our exterior angle because it is
on the outside of our triangle.
Remote interior angles are 2 nonadjacent,
interior angles.
In this picture, angles 2 and angle 3 those
are our remote interior angles.
They are inside and they are not connecting
or touching angle 1.
The triangle exterior angle theorem tells
us that we can add both the remote interior
angles and they will equal the exterior.
So the formula we can use measure of angle
1 will equal the measure of angle 2 + the
measure of angle 3.
So, let's look at a couple of examples.
If we have the measure of angle s is 58 degrees
and angle r is 63 degrees, we need to find
the missing angle measure.
We want to find our exterior angle measure
which is angle UTS.
Exterior tells us that we need to add 63 and
58 (these are our two remote interior angles,
they are inside the triangle and they are
not touching the exterior angle).
63 + 58 = 121 degrees.
So the angle of UTS is 121 degrees.
Last example for today.
We have in this triangle two missing angles
and the angle which forms 58 degrees.
We have vertical angles, which we learned
from previous lessons, and they are 101 degrees
because they are vertical angles.
Based on that information, we can say that
in a triangle, three angles, angle x + 58
+ 101 = 180.
101 + 58 = 159.
so x+ 159 = 180.
We subtract 159 to both sides, and x is going
to equal 21 degrees.
