Question

Write an indirect proof proving only one angle of an obtuse triangle is greater than 90°.

Answer 1

Indirect proof is a proof of contradiction. List the things you know abot an obtuse triangle

- It's a triangle so must have interior angles that total 180º

\[ a+b+c=180º\]

- An obtuse must have one angle that is larger than 90º and less than 180º

\[90º<a<180º \ \ \ \ \mathrm{or} \ \ \ \ 90º<b<180º \ \ \ \ \mathrm{or} \ \ \ \ 90º<c<180º\]

Now you assume that at least 2 angles of the triangle have angles greater than 90º

Can you write an equation or expression for that part?

Answer 2

I don't think I can.

Answer 3

well put somethin' down and we'll see where it takes us ^_^

Answer 4

I BELIEVE IN YOU

Answer 5

where do you get the degrees sign

Answer 6

are you on a mac or a pc?

Answer 7

pc

Answer 8

alt+0176 on the number pad

Answer 9

90 degrees> a> 180 degrees and 90 degrees> b> 180
something like that?

Answer 10

exactly! (well, almost. it's 90º<a<180º and 90º<b<180º - the inequalities open to the right to make it say "90 degrees is less than a, a is less than 180 degrees," but I knew what you meant. The AND was the important part)

So if a AND b are both greater than 90º, what happens to c when we try to sum up the angles

\[(a+b)+c=180\]

Answer 11

c will make the total sum of the degrees over 180.

Answer 12

Exactly!!


If c has any non-negative value then the left hand side of the equation will be greater than 180º


(also, alt+0176 means you hold down the alt key then press 0176 on the number pad, then release the alt key)

Answer 13

You can show that graphically by

Answer 14

I understand that, thank you. But ordering the facts into a logical proof is also giving me a hard time.

Answer 15

^_^ welcome

You start with the things that must be true for every triangle



\[\textrm{For every triangle} \\ \\ \ \\ \ a+b +c\overset{!}{=}180º\]

\[ \textrm{For every obtuse triangle}
\\ \
\\ \ 90º<a<180º\]

\[\textrm{Assumption: both a and b are greater than 90º}
\\ \
\\ \ 90º<a<180º \ \ \ ; \ \ \ 90º<b<180º\]

\[\textrm{If true}
\\ \
\\ \ 180º<a+b<360º\]

\[\textrm{Then for} \\ \ \\ \ (a+b)+c\overset{!}{=}180º
\\ \ \\ \ \\ c<0\]

\[ \textrm{This makes no physical sense. Thus only} \\ \ \\ \ \\ 90º<a<180º \\ \ \\ \ \textrm{can be true. There exists only one obtuse angle in an obtuse triangle. QED.}\]


the equals sign means "must be equal to." You could also use the triple equals sign

\[ \equiv\]

Answer 16

I think I know what to do now. Thanks for your help. :)

Answer 17

welcome ^_^ Good luck with the writeup!