Equilateral and Isosceles Example Problems

Three example problems involving isosceles and equilateral triangles. 2 from Art of Problem Solving (by Richard Ruscyk) book

Let's do some example problems

using our newly acquired knowledge

of isosceles and equilateral triangles

So over here I have a kind of a triangle within a triangle

And we need to figure out this orange angle right over here,

and this blue angle right over here

And we know that side AB is equal to！

or segment AB is equal to segment BC, which is equal to segment CD,

or we could also call that DC

So first of all, we see that triangle！ triangle ABC is isosceles

And because it's isosceles,

the two base angles are going to be congruent

This is one leg, this is the other leg right over there

So the two base angles are going to be congruent

So we know that this angle right over here is also 31 degrees

Well, if we know 2 of the angles in a triangle,

we can always figure out the third angle

They have to add up to 180 degrees

So if we！ we could say 31 degrees + 31 degrees

+ the measure of angle ABC is equal to 180 degrees

You could subtract 62

This right here is 62 degrees

You subtract 62 from both sides,

You get the measure of angle ABC is equal to

Let's see

You subtract another 2, you get 118 degrees

So this angle right over here is 118 degrees

Let me just write it like this

This is 118 degrees

Well, this angle right over here,

this angle right over here is supplementary to that 118 degrees

So that angle + 118 is going to be equal to 180

We already know that that's 62 degrees!

62 + 118 is 180, so this right over here is 62 degrees

Now, this angle is one of the base angles for triangle BCD

I didn't draw it that way, but this side and this side are congruent

BC had the same length as CD

Those are the 2 legs of an isosceles triangle

You can kind of imagine it was turned upside down

This is the vertex

This is one base angle

This is the other base angle

Well, the base angles are going to be congruent

So this is going to be 62 degrees, as well

And then finally, if you wanna figure out this blue angle

The blue angle + these two 62 degree angles

are gonna have to add up to 180 degrees

So you get 62 + 62 + the blue angle,

which is the measure of angle BCD, measure of angle BCD,

is gonna have to be equal to 180 degrees

These 2 characters, let's see

62 + 62 is 124

You subtract 124 from both sides,

you get the measure of angle BCD, is equal to let's see

If you subtract 120 you get 60

and then you have to subtract another 4

So you get 56 degrees

So this is equal to 56 degrees

And we're done!

Now, let's！ we could do either of these

Let's do this one right over here

So what is the measure of angle ABE

So they haven't even drawn segment E here

So let me draw that for us

And so we have to figure out the measure of angle ABE

So we have a bunch of congruent segments here

And, in particular, we see that triangle ABD,

all of its sides are equal

So it's an equilateral triangle,

which means all of the angles are equal

And if all of the angles are equal in a triangle,

they all have to be 60 degrees

They all have to be 60 degrees

So, then, all of these characters are going to be 60 degrees

Well, that's part of angle ABE,

but we have to figure out this other part right over here

And to do that,

we can see that we're actually dealing with an isosceles triangle,

kind of, kind of tipped over to the left

This is the vertex angle

This is one base angle

This is the other base angle

And the vertex angle right here is 90 degrees

And once again, we know that it's isosceles because this side,

segment BD is equal to segment DE

And once again, these two angles + this angle right over here

is going to have to add up to 180 degrees

We call that an x, you call that an x

You get x + x + 90 is going to be 180 degrees

So, you get 2x +, let me just write it down

Don't wanna skip steps here

We have x + x + 90 is going to be equal to 180 degrees

x + x is the same thing as 2x, + 90 is equal to 180

And then we can subtract 90 from both sides

You get 2x is equal to 90, or divide both sides by 2,

you get x is equal to 45

X is equal to 45 degrees

And then we're done,

because angle ABE is going to be equal to

60 degrees + the 45 degrees

So it's going be this whole angle, which is what we care about

Angle ABE is going to 60 + 45 which is 105 degrees

And now we have this last problem over here

This one looks a little bit simpler

I have an isosceles triangle, this leg is equal to that leg

This is the vertex, and we have to figure out B

And the trick is that, wait,

how do I figure out one side of the triangle

if I only know one other side, or I need to know 2 other sides

And we'll do the exact same way we just did

to the 2nd part of that problem

If this is an isosceles triangle, which we know it is

Then this angle is going to be equal to that angle there

And so if we call this x, then this is x as well

We get x + x + 36 degrees is equal to 180

The two x's when you add them up, you get 2x and then

I'll just！ I won't skip steps here

2x + 36 is equal to 180

Subtract 36 from both sides

We get 2x！ that 2 looks a little bit funny

And then you wanna subtract another 6 from 150 gets us to 144

Did I do that right?

Divide both sides by 2, you get x is equal to 72 degrees

So this is equal to 72 degrees

And we are done