Last time we learned about the SAS Theorem, which is used as a shortcut to show that two triangles are congruent. In fact, this is not the only “shortcut” of this type — there are three others, which we will learn about today. They are called angle side angle (ASA), angle angle side (AAS), and side side side (SSS).
This video gives an overview of the congruence theorems above, and also discusses a little bit why the other combinations (AAA and SSA) are not theorems.
Figuring out which theorem applies can sometimes be a challenge — this is usually one step in a larger problem, but the following video focuses just on this step. It presents a collection of triangles and uses them to discuss when each of the theorems can be applied.
In your homework, you will see that we often are not given all the information required — we have to rely on other facts (such as facts about parallel lines, and so on). This video gives an example of how we use these theorems in practice.
Example: Showing that a point is the midpoint of a line segment.
The final topic of the day is isosceles triangles, or triangles with two equal sides. Even though the definition speaks about sides, it is related to angles by the following theorems:
- If two sides of a triangle are equal, then the angles opposite those sides are equal.
- If two angles of a triangle are equal, then the sides opposite those angles are equal.
Why is this true? We can prove it, using what we have learned about congruent triangles:
What about having all three sides equal? Such a triangle is called equilateral. Once again, we can use what we have learned before to show the following:
- If all three sides of a triangle are equal, then all three angles are also equal (“an equilateral triangle is equiangular”)
- If all three angles of a triangle are equal, then all three sides are also equal (“an equiangular triangle is equilateral”)
Many of the homework problems that you will encounter will require you to combine various facts from geometry and also algebra. Here is an example that involves parallel lines and isosceles triangles: