It’s Professor Dave, let’s compare triangles.
In geometry, we are sometimes concerned with proving things.
Unlike the real world, where nothing can ever be proven, only shown to be consistent beyond
reasonable doubt, in the perfect realm of mathematics, things can indeed be proven to
be inarguably true, and our construction of proofs, as meticulous as they may seem, are
incredible demonstrations of logic.
One of the things we will want to prove is the relationship between two shapes, like
two triangles.
Two triangles might be unrelated, or they may have some special relationship.
They could be similar, meaning they have precisely the same shape, just different sizes, or they
could be congruent, meaning they are absolutely identical.
When trying to demonstrate similarity, we have to look at the corresponding sides and
angles.
Here are two triangles that look like they might be similar.
We can prove they are similar if the corresponding sides have lengths that are proportional,
or if the corresponding angles are identical.
This means that if this triangle has sides equal to two, four, and six, while this one
has sides equal to four, eight, and twelve, the two triangles must be similar.
Also, if this triangle has angles of thirty, sixty, and ninety, and so does this one, these
triangles must be similar, no matter what the side lengths.
We can also prove triangle congruence through different methods.
First, there is the side side side theorem.
This states that if the three sides of a triangle are identical to the three sides of the other
triangle, they must be congruent, because there is only one way for three sides of particular
lengths to come together.
If any of the angles were any different, the side lengths would have to change accordingly
to maintain a closed shape.
Similarly, we can use the side angle side theorem.
If two sides of a triangle and the angle between them are identical to the two sides and included
angle of another triangle, they must be congruent.
If we construct this much of a triangle, there will only be one possible way to connect the
two open ends.
A variation of this is the hypotenuse-leg theorem.
This uses the hypotenuse and one leg of a right triangle, as well as the right angle
that is implied.
If two right triangles have the same values for the hypotenuse and one leg, they must
be congruent.
There is also the angle side angle theorem.
If there is a side of a particular length, and the other two sides that will extend from
endpoints of this line segment must emerge at particular angles, then there is only one
place where they could meet and finish the triangle, so any triangles with these two
angles and an included side of this length must be congruent.
Lastly, there is the angle angle side theorem.
This is the same as the angle side angle theorem except that the side involved is not the one
in between the two angles in question, it is one of the other two sides.
This also proves congruence because if these angles are fixed and this side length is fixed,
the other two sides must also be of particular lengths if this is to form a closed triangle.
These are some ways that we can prove triangle similarity and congruence, and once proven,
we can calculate all kinds of information, so with that understood, let’s check comprehension.