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Provided that A and B are acute angles and sin A = sin B, prove that A = B

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,949

Draw right triangles ABC and PQR.

Prove that B = Q

We know that Sin B = Sin Q so:

We can manipulate the above to

b / q = c / r = k

b = kq and c = kr

Pull the k^2 out to simplify.

So

ΔABC is similar to ΔPQR

B = Q

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**{7/3}****Member**- Registered: 2013-02-11
- Posts: 210

Sin A=a/c and sin B=b/c then a=b ,so in two right triangles with angle A and B ,oposite side is a=b,hypottenuse is c,other side must be equal.thus two triangles are congruent,A=B

*Last edited by {7/3} (2013-06-29 16:49:10)*

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Thanks

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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{7/3} wrote:

Sin A=a/c and sin B=b/c then a=b ,so in two right triangles with angle A and B ,oposite side is a=b,hypottenuse is c,other side must be equal.thus two triangles are congruent,A=B

But the two right triangles are not congruent! You certainly cannot say that two right triangles containing equal angles are congruent.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,525

They both have a side 'c'.

Bob

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What if they do not have?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**zetafunc.****Guest**

Are you required to give a geometric proof?

**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,525

Agnishom wrote:

What if they do not have?

Start with two right angled triangles, both with hypotenuse c. Let one have angle A and the other B.

Use the equal sines property to show that a = b. Then they are congruent.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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In that case are we not assuming that A + B = 90degree ?

zetafunc wrote:

Are you required to give a geometric proof?

Well, yeah. But I would love to see any other proofs you have

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,949

I thought that you would have found post #2 particularly relevant.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,525

hi Agnishom,

Let's start again, shall we. Draw a right angled triangle with angle A, hypotenuse c, opposite a. Draw another with hypotenuse c and opposite B. See diagram below.

From the first sinA = a/c

from the second sinB = b/c

But sinA = sinB , therefore a/c = b/c , therefore a = b

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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