Congruence and similarity — Harder example
About this video
Triangle ABC is isosceles with a 100 degree top angle, and it overlaps triangle DEF. Sal uses the equal base angles and the 180 degree angle sum to find each base angle is 40 degrees, then uses a vertical angle and a small triangle with angles p, 40 and 60 to find p = 80.
You'll be able to
- use the fact that an isosceles triangle has congruent base angles.
- use the angle sum of a triangle (180 degrees) to find a missing angle.
- use vertical angles to move an unknown angle into a triangle where I can solve for it.
Key words
- isosceles triangle
- A triangle with two sides of equal length; the angles opposite those sides (base angles) are equal.
- base angles
- The two angles of an isosceles triangle opposite its equal sides.
- interior angles
- The angles inside a triangle; they always add up to 180 degrees.
- vertical angles
- Opposite angles formed by two crossing lines; they are always equal.
Transcript
- [Instructor] In the figure above, triangle ABC is isosceles. So that's triangle ABC. So that big triangle, that's isosceles, which means two of its sides are congruent, have the same length. That is side AB, they explain it here, side AB, so side AB here is equal in length to side BC.
So these two sides are equal, equal in length. So we could say that that side is equal to that side. Actually, I wanna be careful, just because that looks like I'm saying that from that point to that point, but those two blue sides are gonna be equal in length.
Triangle DEF, DEF, overlaps with triangle ABC. Yeah, we see that, they kinda form this Star of David, or kinda this skewed Star of David. What is the value of angle p here? All right, let's work through this. So ABC is an isosceles triangle, and I'm gonna draw it separately here.
So ABC is an isosceles triangle. This is an isosceles triangle. And an isosceles triangle if these two, if these two sides are congruent, then these base angles are going to be congruent. And they already tell us that this angle up here, they already tell us that this angle up here is 100 degrees, and we know that all the interior angles of a triangle add up to 180 degrees.
So if we call, if we call this x degrees, and then this is x degrees, then we see that x plus x plus 100, plus 100, is going to need to be equal to 180. Or we can get that two x, two x, and then if we subtract 100 from both sides, is equal to 80, or that x is equal to 40 degrees.
So just like that, we're able to figure out that both of these, both of these are going to be equal to 40 degrees. And that came straight out of the fact that ABC is an isosceles triangle. These two base angles are going to be congruent. So this is 40 degrees and this is 40 degrees.
Now, now let's go to this little small triangle right over here, this little small triangle right over here. Once again, we know that the interior angles of a triangle add up to 180 degrees. So if this right over here, actually this angle right over here is going to be the same as p, because it's vertical.
It's a vertical angle with p. So this is going to be p degrees as well. And now, we see p plus 40, p plus 40 plus 60, plus 60, is going to add up to 180 degrees. That's the interior angles of this little small triangle right over here. Or, we could write p plus, if we add the 40 and the 60, you get 100, p plus 100 is equal to 180.
If we subtract 100 from both sides, we get p is equal to 80 degrees. So what is the value of p, or I guess this is p degrees? So the value of p is, the value of p is 80.
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