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### Course: Class 9>Unit 13

Lesson 2: Triangles 7.2

# Proofs concerning isosceles triangles

Sal proves that the base angles in isosceles triangles are congruent, and conversely, that triangles with congruent base angles are isosceles. He also proves that the perpendicular to the base of an isosceles triangle bisects it. Created by Sal Khan.

## Want to join the conversation?

• What if I solve this by saying that Triangle ABC is congruent to itself (through SAS) in this way - 1. AC congruent to AB (Symmetric Property)
2. Angle A congruent to Angle A (Reflexive)
3. Triangle ABC congruent to Triangle ABC (SAS)
4. So Angle B congruent to Angle C (CPCTC)
Is this an acceptable way of proving it?
• Yes, that is a very good strategy indeed. That is called Pappus' proof, because Euclid didn't think of it when he wrote The Elements. (Euclid didn't use Sal's proof either, because this is Proposition 5 and SSS doesn't get proved until Proposition 8.) It's much simpler once you get over the initial hump of how weird it is to have a congruence proof with what looks like only one triangle.

There is one change you should make to the proof, though. You should make your point in step 3 more clear by saying that Triangle ABC is congruent to triangle ACB -- you see how I lined up the letters to make it clear what the corresponding vertices are? Because, to be honest, you could show that any triangle is congruent to itself whether it is isosceles or not.
• At , Sal says that we have a lot of triangle congruency theorums to use. But in earlier videos, Sal calls them postulates because they can't be proven. Which one is right???
• They are all theorems. To be precise, SAS is Proposition 4, SSS is Proposition 8, and ASA and AAS are combined into Proposition 26. Sal may have been thinking that Euclid never formally defined what he meant by congruence.
• What is the difference between congruency and equality?
• Equality (=) is used for measurements: length (inches, cm, etc.), angle measures (degrees), area (cm^2)

Congruency (= with a ~ over it) is used for objects: line segments, angles, polygons, circles. These are things that have the all the same properties (equal measurements.) Orientation is not important, so you can rotate or flip a polygon and it can still be congruent. But, these objects are not numbers.

Much like the height of a book shelf can be measured, say 4 feet. But the book shelf would not be congruent to another 4 foot tall bookshelf unless it was equal in all measurements and qualities (height, length, width, number of shelves, color, etc.)
• How exactly would you write a proof? Like if you were doing a test, would you just write something like Sal writes or would you have to explain it in words?
• Well if you were explaining it to a teacher you can draw while telling your teacher but if not, then you would probably have to write it out because Sal was drawing while telling us.
• "we-we-we"
• so how does this help me with taxes?
• its geometry, it doesn't help with taxes
• Why is that AD is congruent to AD? What is that property called?
• We can call that property the reflexive property. It just says an object must be equal to itself, like 1=1.
• What if you make an angle bisector for angle BAC (the angle bisector would meet segment BC at point D)? Then, angle BAD and angle CAD would be congruent, and you could use AAS to prove that the two triangles are congruent, right? The first pair of congruent angles would be ACD and ABD, the second pair would be angle BAD and angle CAD, and the congruent side would be AD (AD is congruent to AD by the reflexive property). Wouldn't this work as well?
• Yes, this can also work. You are using AAS, and your explanation does make sense.
• what did at sign that sal made mean? at ?
• It means it is congruent
• At he says that on and isosceles triangle if you have two angles that are the same then the line that connects them is a congruent line. Is this true with other kinds of triangles?
• No, because the whole proof starts with the fact that there are two equal sides, ie it's an isosceles triangle