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## Class 10 (Foundation)

### Unit 9: Lesson 3

Proof of triangle properties

# 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?

• 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.
• 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.
• what did at sign that sal made mean? at ?
• 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
• In he says AB is congruent to segment AC but he wrote AC before AB
• In Geometry the order only matters when you say AC and AB individually. a is congruent to a, while C is congruent to B. saying CA is congruent to AB would be incorrect. Tell me if it is still unclear, I'm not the best explainer
• 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?
• i think you are right it is hard to understand step3
• can I do the second proof (at around ) by construction? that is, if I take a compass and can make a circle with center A and radii AB and AC, would this be a way of proving the statement AC = AB? I assume so, since all radii in a circle are equal in length to each other, right?
• Yes, you can. Technically, you can construct any triangle, and it really helps me figure out difficult geometric equations. Also, constructing a triangle helps show your work so you can refer to it later.
• what do the base angles have to do with this
• It's something that we will need to know eventually.
(1 vote)
• At Sal says that we can always construct an altitude of a triangle, how does he know this? How can he be so sure that you'll always be able to have a line drawn from the vertex always be perpendicular to one of the triangle sides? Or one of the sides of the triangle extended? (Since an altitude can be defined in both ways I stated.)