Question

I would really appreciate anybody who helps me im not asking to answer it or give me the answer just for help. Here is my question- Write an indirect proof proving only one angle of an obtuse triangle is greater than 90 degrees.

Answer 1

Just so we're clear on terminology, an indirect proof is a proof in which you prove that a statement is true by first assuming that its opposite is true. If this assumption leads to an
impossibility, then you have proved that the original statement is true.

So given an obtuse triangle ABC, you want to prove that only one angle can be >90 degrees

Lets start with the opposite of what we're trying to prove:

suppose angle A>90 degrees and angle B>90 degrees (this would mean more than one angle is obtuse)

the sum of all angles of a triangle must be 180 degrees
A+B+C = 180 (triangle sum theorem)
if A>90 and B>90, C would have to be negative
negative angles cannot exist, therefore no more than one angle in a triangle can measure >90 degrees

(source/cleaner version: http://www.nexuslearning.net/books/ml-geometry/Chapter5/ML%20Geometry%205-6%20Indirect%20Proof%20and%20Inequalities%20in%20Two%20Triangles.pdf )

Answer 2

Yes

Answer 3

Ok thanks :)

Answer 4

Im understanding a little bit more

Answer 5

I still need a little more help please...

Answer 6

well... what dont you understand now?

Answer 7

Im re-reading everthing over again

Answer 8

ok. you can tell me what you dont understand. OR if you really dont get it, we can start all over from the very beginning

Answer 9

ok thanks

Answer 10

Ok can we just start from the beginning cause im still confused.

Answer 11

ok. lets make a picture for this

lets make obtuse triangle ABC with obtuse <B

Answer 12

So angle B=90 degrees

Answer 13

so we're trying to prove that only <B is greater than 90 in this obtuse triangle

Answer 14

Oh ok

Answer 15

so could say that^ but that would be a later step.
there actually different ways to prove this

Answer 16

so im confused does all the angles equal 180 or 360

Answer 17

180 because you have a triangle. all 3 angles in any triangle add up to 180

360 is for quadrilaterals or 4 sides shapes

Answer 18

oh ok cause i kept reading 360 then 180 so i got confused

Answer 19

ok. how do you want to start it.
there are two ways
1) <B=90 which is what you said
or
2) both <A and <B are greater than 90

Answer 20

1

Answer 21

ok. so your first step would be to state the given. which would be that triangle ABC is an obtuse triangle

Answer 22

Ok

Answer 23

then you would say <B=90. im not sure what your teacher told you to put as the reason,
but my teacher calls it the supposition of an indirect proof or "supp. of IP" for short

<B=90 is your "assumption" which is always going to be the opposite of what you're trying to prove

Answer 24

She didn't tell me anything i can do it anyway.

Answer 25

ok. but thats what you do in an indirect proof.
state the opposite of what you're suppose to prove(your assumption)
and keep going until you have something that contracts what you had "assumed"

Answer 26

Ok thanks :)

Answer 27

ok. now remember what the sum of all three angles in a triangle add up to?

Answer 28

180 :)

Answer 29

yes!
so now we can say
<A + <B + <C= 180 , right?

this equation means that all three angles of this triangle add up to 180

Answer 30

That's right :)

Answer 31

ok. now we use substitution. substitute <B=90 into that equation. what do you get?

Answer 32

angle A+90+b=180?

Answer 33

close.

<A + 90 + <C =180

Answer 34

Do i have to switch it?

Answer 35

no. you substituted the value in