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### Course: High school geometry > Unit 3

Lesson 5: Working with triangles# Finding angles in isosceles triangles

The measures of two angles of an isosceles triangle are 3x+5 and x+16. Find all possible values of x. Created by Sal Khan.

## Want to join the conversation?

- What is an isosceles triangle?(10 votes)
- An
**isosceles**triangle is a triangle which has*at least*to sides equal to each other. Notice that all equilateral triangles are isosceles.(32 votes)

- can we write an angle in decimal form? like measure of angle ABC = 95.55(6 votes)
- Yes you can, but they are usually written in degrees-minutes-seconds.

so 95.55 = 95º 33' (ninty five degrees, thirty three minutes).

or 95.44444 = 95º 26' 39.98" (Here, after seconds we write decimals...)

However, I only do this because my calculator has a key for doing it easely, on other situations using decimals should be perfectly fine.(5 votes)

- can u do questions that ask find the x of an iscoseles triangle and with a given angle?(4 votes)
- The sum of angles in a triangle are 180, and if you have an iscoseles triangle, the angles opposite the congruent sides are congruent also. So if you have x as one of the angles opposite and a vertex angle of x + 30, the other opposite angle is also x, so x + x + x+ 30 = 180 which you can solve.(6 votes)

- how do you know if you use the number twice in the equation or the x??(4 votes)
- In an isosceles triangle, there are two base angles and one other angle. The two base angles are equal to each other. So say you have an isosceles triangle, where only two sides of that triangle are equal to each other. And then you have 36 degrees as one of your base angles. The other base angle will equal 36 degrees too.

36 + 36 + x = 180 degrees

36 + 36 = 72

72 + x = 180

180 - 72 = 108

x = 108.

REMEMBER: THIS ONLY WORKS IF YOU HAVE TWO BASE ANGLES. IN THIS CASE, WE HAD TO USE THE NUMBER TWICE (BECAUSE OF THE BASE ANGLES) AND X (DENOTE THE UNKNOWN NUMBER). I REALIZE YOU PROBABLY ALREADY FOUND HELP ANSWERING YOUR QUESTION; BUT IF NOT, HERE IT IS.(4 votes)

- Is there a video explaining the hinge theorem? I need to see it.(4 votes)
- Look at this image, it helps: http://cdn.virtualnerd.com/tutorials_json/Geo_05_01_0004/assets/Geo_05_01_0004_D_01_06.png

The hinge theorem states that if two sides of two corresponding triangles are congruent (XZ = MP and ZY = PN) then whichever included angle is larger, means the opposite is larger. Since 57 > 55, then the side opposite to 57 is larger than the side opposite to 55(3 votes)

- What does congruent mean?(3 votes)
- Two shapes are congruent if they have the same shape and size. Or, any shape that can be translated, rotated, or reflected to match another shape means that those two shapes are congruent.(4 votes)

- Can a triangle have negitive angles or is that impossible?(2 votes)
- In geometry, all angles measurements are positive, at least in the geometry we are studying on this site.

In Trigonometry (after you finish with the simplest levels--trigonometry of right triangles) and also in Calculus and Physics and when studying vectors, and in other similar maths, angles are just as likely to be positive as negative because`positive angles rotate counterclockwise`

and`negative angles rotate clockwise`

.

Even more mind-blowing is that lengths can be positive or negative depending on their direction and location relative to the starting point. This allows us to describe the motion of rockets and speeding atomic particles and many other interesting things.(6 votes)

- What is congruent and what is it used for?(3 votes)
- You can set up numbers or variables as equal such as 6=6 or x=6 or 3x+2=8. Congruent is used for figures which have the same shape and size. If you are building, you often want a bunch of boards that are all congruent. Machines make parts that are all congruent so that you can replace them. In math, if two figures are congruent, we can solve for parts of the figure because we know that congruent sides/angles have equal measures.(4 votes)

- Would it make sense to make 2 hypothetical equilateral triangles, one with the angle measures ‘3x+5’ and the other with ‘x+16’, if we consider that all equilateral triangles are also isosceles?(2 votes)
- No because equilateral triangles have all angles equal to 60 degrees. Setting 3x+5=60 gives x=55/3 and setting x+16=60 gives x=44. You should not have two different values of x.(5 votes)

- How can you tell right off the bat if it is an isosceles triangle?(3 votes)
- If the triangle has two sides that have the congruent mark on them, you know that it's isosceles. If you have the lengths of all three sides of a triangle and two of them are the same, it's an isosceles triangle. Another way would be to use the Converse of the Isosceles Triangle Theorem. The Converse of the Isosceles Triangle Theorem states that if two base angles, or two angles that are consecutive, are congruent, then the triangle must be congruent.(3 votes)

## Video transcript

The measure of two angles
of an isosceles triangle are 3x plus 5 degrees, we'll
say, and x plus 16 degrees. Find all possible values of x. So let's think about this. Let's draw ourselves an
isosceles triangle or two. So it's an isosceles triangle,
like that and like that. And actually, let me
draw a couple of them just because we want
to think about all of the different
possibilities here. So we know, from what we know
about isosceles triangles, that the base angles are
going to be congruent. So that angle is going to
be equal to that angle. That angle is going to
be equal to that angle. And so what could the 3x plus
5 degrees and the x plus 16, what could they be measures of? Well, maybe this
one right over here has a measure of
3x plus 5 degrees. And the vertex is the other one. So maybe this one up here
is the x plus 16 degrees. The other possibility
is that this is describing both base
angles, in which case, they would be equal. So maybe this one is 3x plus
5, and maybe this one over here is x plus 16. And then the final
possibility-- actually we haven't exhausted
all of them-- is if we swap these two--
if this one is x plus 16, and that one is 3x plus 5. So let me draw ourselves
another triangle. And obviously swapping
these two aren't going to make a
difference because they are equal to each other. And then we could make that
one equal to 3x plus 5. But that's not going to
change anything either because they're
equal to each other. So the last situation is
where this angle down here is x plus 16, and this
angle up here is 3x plus 5. This is 3x plus 5. So let's just work
through each of these. So in this situation, if
this base angle is 3x plus 5, so is this base angle. And then we know that
all three of these are going to have to
add up to 180 degrees. So we get 3x plus
5 plus 3x plus 5 plus x plus 16 is going to
be equal to 180 degrees. We have 3x. Let's just add up. You have 3x plus
3x, which gives you 6x, plus another x gives you 7x. And then you have 5
plus 5, which is 10, plus 16 is equal to 26. And that is going
to be equal to 180. And then we have, let's
see, 180 minus 26. If we subtract 26 from both
sides, we get 180 minus 20 is 160, minus another 6 is 154. You have 7x is equal to 154. And let's see how many times--
if we divide both sides by 7, 7 will go into 140 20 times,
and then you have another 14. So it looks like it's 22 times. So x is equal to 22. Is that right? 20 times 7 is 140. 140 plus 14 is 154. So we have x is equal
to 22 degrees in this the first scenario. Now let's think about the
second scenario over here. Now we have these
two characters are going to be equal to each
other because they're both the base angles. So you have 3x plus 5
is equal to x plus 16. Well, you can subtract
x from both sides. And so this becomes 2x
plus 5 is equal to 16. We can subtract
5 from both sides and you get 2x is equal to 11. And then you can
divide both sides by 2, and you get x is equal to 11/2. So that is our second scenario. And then we do our third
scenario right over here. If this base angle is x plus
16, then this base angle right over here is also
going to be x plus 16. They are congruent. And then we can
do the same thing that we did for
the first scenario. All of these angles are going to
have to add up to 180 degrees. So we have x plus 16 plus
x plus 16 plus 3x plus 5. When you add them
all together, you're going to get 180 degrees. Now let's add up
all the x terms. x plus x is 2x plus 3x is 5x. So we get 5x. And then you have
16 plus 16 is 32. 32 plus 5 is 37. Plus 37 is equal to 180 degrees. Subtract 37 from
both sides, and we get 5x is equal to
180 minus 30 is 150. So that gets us to 143. So it's not going
to divide nicely. Divide both sides by 5,
you get x is equal 143/5, which we can just leave
as an improper fraction. You could write it
as a mixed number or however else you
might want to write it. And we're done. These are the three
possible values of x, given the information that
they gave us right up there.