Indirect Proof

jtayag0622

New member
Joined
Sep 10, 2018
Messages
41
Hi guys,
I tried to write down an indirect proof in paragraph form for this:
indirect proof smaple.jpg
This is what I did so far:

Assume temporarily that ∠Y is not acute, so it’s either obtuse, right or straight angle. We know that the sum of the interior angles of angle.

Case 1: ∠Y is obtuse
We know that a triangle can only have one obtuse angle, so this contradicts the given information that m∠X = 100 and x is obtuse. So this assumption must be false.

Case 2: ∠Y is right angle or straight angle
X + Y + Z = 180
90+100+z =180
Z = -10
180+100 +z= 180
Z = -100

This contradicts the fact shapes cannot have negative angle measure. So this assumption must be false.
Both assumptions ended with a contradiction…

I didn't continue it since I think I'm lost so please help me :)
 
Hi guys,
I tried to write down an indirect proof in paragraph form for this:
View attachment 10302
Question: Are you required to us an indirect proof?
If not, then there is a point \(\displaystyle T\) such that \(\displaystyle T-X-Y\), i.e. X is between T & Y.

Now we know that \(\displaystyle m(\angle ZXT)=m(\angle Z)+m(\angle Y)\). WHY?

What dose that tell us about \(\displaystyle m(\angle Y)~?\)
 
Last edited:
Hi guys,
I tried to write down an indirect proof in paragraph form for this:
View attachment 10302
This is what I did so far:

Assume temporarily that ∠Y is not acute, so it’s either obtuse, right or straight angle. We know that the sum of the interior angles of angle.

Case 1: ∠Y is obtuse
We know that a triangle can only have one obtuse angle, so this contradicts the given information that m∠X = 100 and x is obtuse. So this assumption must be false.

Case 2: ∠Y is right angle or straight angle
X + Y + Z = 180
90+100+z =180
Z = -10
180+100 +z= 180
Z = -100

This contradicts the fact shapes cannot have negative angle measure. So this assumption must be false.
Both assumptions ended with a contradiction…

I didn't continue it since I think I'm lost so please help me :)

I don't think you need two cases; just assume that Y >= 90, which is the negation of Y being acute. You can just do something like what you did for case 2 (but with an inequality), for this assumption.

But why do you think you're lost rather than finished? What do you think is missing? If your work is correct (though not what I would do), then you've shown that the assumption leads to a contradiction, so it must be false, and Y must be acute.
 
I don't think you need two cases; just assume that Y >= 90, which is the negation of Y being acute. You can just do something like what you did for case 2 (but with an inequality), for this assumption.

But why do you think you're lost rather than finished? What do you think is missing? If your work is correct (though not what I would do), then you've shown that the assumption leads to a contradiction, so it must be false, and Y must be acute.

Well, I actually thought of using Y≥90 . But I don't know how to explain it and how to make the solution for the inequality with the 180°(Triangle Sum Theorem).

I thought I was lost because I thought my steps seemed a bit wrong. Hehe . Anyways, thanks !
 
Well, I actually thought of using Y≥90 . But I don't know how to explain it and how to make the solution for the inequality with the 180°(Triangle Sum Theorem).

I thought I was lost because I thought my steps seemed a bit wrong. Hehe . Anyways, thanks !

Yes, your steps could be improved; but it's also good to see what it correct about what you did!

Given X = 100, you have

X + Y + Z = 180, so

Z = 180 - 100 - Y = 80 - Y.

If Y >= 90, what must be true of Z?
 
Yes, your steps could be improved; but it's also good to see what it correct about what you did!

Given X = 100, you have

X + Y + Z = 180, so

Z = 180 - 100 - Y = 80 - Y.

If Y >= 90, what must be true of Z?

...that Z would be negative. But that wouldn't be possible since angle measures of polygons should be positive.
Therefore the temporary assumption that Y≥90 must be false. It follows that Y is an acute angle.

Is my conclusion right?
 
...that Z would be negative. But that wouldn't be possible since angle measures of polygons should be positive.
Therefore the temporary assumption that Y≥90 must be false. It follows that Y is an acute angle.

Is my conclusion right?

Correct.

You could do a little more with the inequality, though. Given that Y >= 90, you know that -Y <= -90; so if Z = 80 - Y, then Z <= 80 - 90 = -10.

It's enough, of course, just to know it would have to be negative.
 
Correct.

You could do a little more with the inequality, though. Given that Y >= 90, you know that -Y <= -90; so if Z = 80 - Y, then Z <= 80 - 90 = -10.

It's enough, of course, just to know it would have to be negative.

Thank you so much! I don't know how to give back your help since you're a lot better than me in Math. Thank you again! I really do appreciate it :)
 
Top