Main content

## Geometry (all content)

### Course: Geometry (all content) > Unit 2

Lesson 7: Angles between intersecting lines- Angles, parallel lines, & transversals
- Parallel & perpendicular lines
- Missing angles with a transversal
- Angle relationships with parallel lines
- Parallel lines & corresponding angles proof
- Missing angles (CA geometry)
- Proving angles are congruent
- Proofs with transformations
- Line and angle proofs

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Proving angles are congruent

Sal proves that two angles are congruent in a really interesting triangle like figure.

## Want to join the conversation?

- It may seem like a proof for corresponding angles formed by an intersection of two parallel lines are equal. However, as far as I remember, we use corresponding angles are equal to prove that the sum of angles in a triangle is equal to 180 (the fact that is used in this video). Is it a vicious circle?)(23 votes)
- Both propositions can be proved by
*Euclid's fifth postulate*, an axiom in Euclidean geometry.(1 vote)

- Couldn't you prove this faster by saying MLK and NLJ are similar, as they have two angles with the same measure (C and the 90° angle), and all angles in similar triangles are equal?(6 votes)
- yes but that is what sal is trying to explain in this video. It as all nice that you can say that as they have two angles with the same measure and all angles in similar triangles are equal but sal was showing mathematically why you are able to say that.(4 votes)

- why do you have to prove they are congruent?(5 votes)
- So that you can tell if they are congruent or not, before moving on the the new step in geometry.(3 votes)

- Can't you just use alternate interior angles which would be a lot simpler?(4 votes)
- That's a good strategy indeed:D.

why i havent thought this earlier?*-*(1 vote)

- Forgive me if this is specified within the video, but how did Sal know that all of the interior angles equal 180 when put together in a triangle?(2 votes)
- The sum of the angles of a triangle is equal to 180°.(4 votes)

- Is there a way to find the actual degrees of angles a and c without using a protractor?(2 votes)
- In trigonometry, you can find angle measures based on how steep they are. Every steepness has a different angle associated with it.(3 votes)

- wouldn't an easier proof be to simply say that both triangles share angle c and both have a 90-degree angle, meaning the final angle for both has to be 180-90-c?(2 votes)
- Yes, and that is essentially what is shown in this video, though explained more thoroughly.(2 votes)

- so the sides of a right triangle equal 90 degrees, and the angles equal 180?(2 votes)
- Several issues with your question, sides of a triangle are never measured in degrees, they are measured in units (including units, feet, meters, etc.) and there is no specific measurement unless you are given a particular right triangle. The only thing for sure about the sides of a right triangle is that the Pythagorean Theorem (a^2 + b^2 = c^2) will always work.

The two acute angles of a right triangle are complementary (add to be 90 degrees) which does mean that there are 180 degrees in a right triangle, but this can be generalized to any triangle.(2 votes)

- I don't really understand the point of this video. We already know that vertical angles in transversals are also congruent but what was the point of this video? To prove that a=b and how?(2 votes)
- The point of the video is about Sal proving that two angles are congruent in a really interesting triangle like figure.(2 votes)

- how do you do a negative parallel problem? none of us can figure it out.(2 votes)
- What do you mean by that? Are you simply asking how to find parallel lines that have negative slopes?(2 votes)

## Video transcript

We have an interesting looking
diagram here. Let's see if we know a few things
about this diagram. Let's say we know that line MK is parallel to line NJ. So this line is parallel to this line. This is line MK, this is line NJ. Now, given that
and all the other information on this diagram, I'm hoping to prove that the measure
of this angle LMK is equal to the measure
of this angle over here and this angle is LNJ. Another way of writing this is; the measure of LMK is b
and the measure of LNK is a. So we want to prove that b is equal to a using all this information
that we know. Like always I encourage you
to try this on your own before I walk through it. Alright, let's walk through it. The first thing you might see is
I have a triangle formed up here, triangle MLK. What do we know about the measurement
of the interior angles of a triangle? The measures of the interior angles
of a triangle are going to add up to 180 degrees. We know that b,
which is the measure of this angle plus the measure of this angle, c plus the measure of this right angle,
which is plus 90 degrees is going to be equal to 180 degrees. And so if we subtract 90 degrees
from both sides we're going to get b plus c
is equal to 180 degrees minus 90 degrees. It's going to be 90 degrees. Or if we wanted to solve
explicitly for b we could subtract c from both sides, and we could write b is equal to
90 degrees minus c. So that's interesting,
that's one way of expressing b. Can we express a in a similar way? Once again, if at any point
you get inspired I encourage you to do that. If we look carefully we see that
we have triangle NLJ, this really big triangle,
it's really most of the diagram. What's interesting about NLJ
is that J is another right triangle, c is one of the measures
of one of the interior angles, and a is a measure of
the other interior angle. We can write something very similar, we can write a plus c
plus 90 degrees is going to be equal
to 180 degrees. So what can we do here? We can do the exact same process
to solve for a. If we subtract 90 from both sides
and we subtract c from both sides what do we get? We get a is equal to 90 degrees
and if you subtract c from both sides you're going to get 90 degrees minus c. Now this is interesting, b is equal to 90 degrees minus c
and a is equal to 90 degrees minus c. So 90 degrees minus c is equal to a,
it's also equal to b. Or, we can now say that
a must be equal to b, that the measure of angle LMK
which is b is equal to the measure of angle LNJ
which is equal to a.