Today we're going to talk about angle-angle similarity, and we're going to talk about how to prove that two triangles are congruent. You'll recall from the last couple of days, the only way that polygons are congruent, or not congruent, but similar, excuse me, the only way polygons are similar is if their corresponding angles are congruent and their corresponding sides are proportional. Well, today what we're going to do is we're going to narrow that down a little bit just to triangles and say that with triangles, one way I can show that two triangles are congruent is if they're not congruent.
triangles are similar is by showing that just their corresponding angles are congruent. And not only that, but we don't need all three pairs of corresponding angles congruent. We only need two pairs.
Just two pairs of corresponding angles being congruent is enough to prove that triangles are similar. Why do we only need two pairs? Well, think about this for just a half a second.
If angle A and angle D are congruent here, and angle B is congruent to angle E, then clearly angle C and angle F must also be congruent because those triangles must both add up to 180. It's called third angle theorem. We learned it earlier in the year. So that's why two angles are enough. So if I can show that two angles of one triangle are congruent to two angles of another triangle, that proves the triangles are similar. Do I have similar triangles in this picture?
If so, name the similarity. Well, I obviously have angle D congruent to angle G. What I don't have is the other two angles, because C is 26 and H is 64. They aren't congruent. So you might be tempted to say, nope. But I do know triangle sum.
I know that 90 plus 64 plus the measure of angle K has to equal 180. So that's it. 180 is 154 plus, we'll just call it K for now, and I subtract the 154, and I get the measure of angle K equals 26 degrees. Aha.
So measure of angle, so angle K is congruent to angle C. That's two pairs of angles. That's enough.
And I can say triangle D. CE is similar to triangle. That was a terrible similar to sign. Let me try that again. Is similar to triangle, now corresponding order.
G is the right angle. C goes with KH. And that would be my similarity statement.
Reason, explaining your reasoning. Angle, angle, similarity. How about here? Are ABE and ACD similar?
Well, ABE is right here. ACD is this right here. They both have a 52-degree angle. So I know that this is congruent to that, so I have a second angle. Well, I don't know about these two angles, but what's the third angle in both triangles?
They both have angle A, don't they? It's a reflexive angle. Just like we had reflexive sides before, that's a reflexive angle because it's the same angle in both triangles.
Sometimes to help when we get overlapping triangles like that, what we'll do is we'll do this. We'll redraw it. A, B. E and A, C, D. But note that the angle A is the same angle A in both triangles. And this is a 52. And this is also a 52. So absolutely yes, those are similar by angle-angle similarity. Letter B. Do I have enough information here?
Please note what we're given. and that will be your first question of the day. Show that the triangles are similar, write a similarity statement.
Number one, are they similar? Absolutely, they're both equiangular triangles. All the angles are congruent, and the similarity statement can go in any order you want it to, pretty much, because the angles are interchangeable.
Number two is a little more interesting. I have a 32 degree angle, a 58 degree angle. and a right angle of an altitude, do I know that triangles are similar, specifically CDF and DEF? Now, this is a little confusing, so I'm going to separate them, okay?
I'm going to separate them. C here is 32. This is a right angle. So that means that this angle up here must be 58 degrees.
This is D and this is F. This angle right here is 58 degrees. That's angle E. This is still a right angle because both angles are right angles.
And so this angle must be 32 degrees, and that's D. So here, the tricky part is angle C in the triangle here on the left. corresponds to angle D in the triangle on the right. So it's not the same angle D.
Angle D in the triangle left is not the same as angle D in the triangle on the right. But then this angle D, 58, corresponds to this angle E. D goes with E. And finally, F does correspond with F because they're both the right angles. And sure enough, that is angle-angle similarity for those two triangles.
How do we apply this? It's pretty straightforward. We prove the two triangles are similar and then we go ahead and set up and solve proportions just the way we've been doing the last couple days.
Here it says a flagpole casts a shadow that's 50 feet long. At the same time a woman standing nearby is five feet four inches tall, casts a shadow that's 40 inches long. How tall is the flagpole to the nearest foot?
So essentially we have two triangles here. There's first triangle, there's my second triangle. Do we have angle-angle similarity? Well, I'm assuming that both are standing upright, so we'll assume that those are right angles. And we'll assume that since the sun is a point source of light 93 million miles away, that both are going to be, because they are by each other, that the angle to the sun will be exactly the same.
Okay, I can't draw a scaled Sun 93 million miles away on this scale. I'd probably be drawing it well into orbit. So yeah, we'll go ahead and assume that these two angles are the same.
Since we have angle-angle similarity, I can now set up my two triangles. I have this triangle that has 5 feet 4 inches. and 40 inches on this side and this triangle that has 50 feet and there's my unknown right there. Now the only thing that's going to make this interesting is we are going to have to make sure that our units become somewhat consistent. I mean I can do inches and inches here and feet and feet here it'll work out but I can't mix inches and feet in one measurement here.
So I have to do one of two things. I could either convert everything into feet So 5 feet 4 inches is 5.33 feet, and 40 inches is 3 feet 4 inches, or 3.33 feet. And then I could set up my proportion. 5.33 over the corresponding side over here, which is 50, equals 3.33 over question mark, cross multiply, and solve. and that shouldn't be a very long difficult process at all.
In fact, I don't have my calculator with me, but I'm going to go grab it, so bear with me. We cross multiply, we take 50 times 3.33, that's 166.5, and divide that by 5.33, and it gives me approximately 31.2 feet for the shadow. Correction, I just realized I misread the problem.
The shadow is 50 feet long, not the height of the flagpole. So my 50 should actually be over here with the shadow. Small error.
Apologize for that. So 50 times 5.33 and divide that by 3.33, 80 feet long. The height is 80 feet. I was a little confused when I got that answer.
It says, why would I want to know the length of the shadow? I could just measure that. That's on the ground.
Yeah, the height of the flagpole makes more sense to be the unknown here. Okay? But we set up and solve our proportion. Now it's your turn. A child who is 52 inches tall is standing next to the woman in example 3. How long is the child's shadow?
Notice the woman is 5 feet 4 inches, casts a 40-inch shadow. This time I'm giving you the height of the child. You need to draw the triangles and solve for the unknown. And that's it for today.
We'll see you in class.