SSA can prove of the congruence of two triangles?

Hey folks! I recently made a project about the congruence of triangles in an advanced way. The one of the topic is SSA (Side-side-angle). Teachers always say that SSA cannot prove two triangles are congruent (the explanation is here), but I want to tell you guys about SSA can prove the congruence of two triangles with some conditions. Let's check this out!

At first, let a triangle be $\Delta ABC$ such that we only know about the length of $AB, BC$ and the size of $\angle A$ , by the sine formula, we can show that

$\dfrac { BC }{ \sin { \angle A } } =\dfrac { AB }{ \sin { \angle C } } \rightarrow \sin { \angle C } = \dfrac { AB\sin { \angle A } }{ BC }$

From here there are two cases:

Case $1$ , $\sin { \angle C }=1$

There are only $1$ possibility of $\angle C$ , which is $90\degree$ . Then, we know the size of $\angle A, \angle C$ and the length of $AB$ , so we can show that there are only $1$ possible triangle by applying $AAS$ . Therefore, we can conclude that when $\sin { \angle C }=\dfrac { AB\sin { \angle A } }{ BC }=1$ , which means $\dfrac { BC }{ AB }=\sin { \angle A }$ , $SSA$ is true.

Case $2$ , $0<\sin { \angle C }<1$

Then, there are $2$ possibilities of $\angle C$ , $\sin ^{ -1 }{ \left(\dfrac { AB\sin { \angle A } }{ BC }\right) }$ and $180 \degree-\sin ^{ -1 }{ \left(\dfrac { AB\sin { \angle A } }{ BC }\right) }$ . Hence, we need to consider $\angle A$ to find out more conditions of proving $SSA$ . By the angle sum of triangle, $\angle C=180\degree-\angle A-\angle B<180\degree-\angle A$ . We will consider two cases about $\angle A$

Case $A$ , $\angle A \ge 90\degree$

$\angle C<180\degree-\angle A\le 90\degree$ . Then, $\angle C$ can only be $\sin ^{ -1 }{ \left(\dfrac { AB\sin { \angle A } }{ BC }\right) }$ , so there are only $1$ possible triangle similarly. Therefore, we can conclude that when $\angle A \ge 90\degree$ , $SSA$ is true.

Do you notice that? When $\angle A = 90\degree$ , it is the same case as RHS (Right angle-hypotenuse-side) or HL (Hypotenuse-leg)

Case $B$ , $\angle A < 90\degree$

$\angle C<180\degree-\angle A$ . It is true when $\angle C=\sin ^{ -1 }{ \left(\dfrac { AB\sin { \angle A } }{ BC }\right) }$ . Because of we want to have $1$ possibility of $\angle C$ , we need to ignore the other possibility of $\angle C$ , which we get the inequality below:

$90\degree>180\degree-\sin ^{ -1 }{ \left( \dfrac { AB\sin { \angle A } }{ BC } \right) } \ge 180\degree-\angle A\\ \\ \sin { 90\degree } >\sin { \left( 180\degree-\sin ^{ -1 }{ \left( \dfrac { AB\sin { \angle A } }{ BC } \right) } \right) } \ge \sin { \left( 180\degree-\angle A \right) } \\ \\ \dfrac { AB\sin { \angle A } }{ BC } \ge \sin { \angle A } \\ \\ AB\ge BC$

Therefore, we can conclude that when $AB\ge BC$ , $SSA$ is true.

To sum up, there is a triangle $\Delta ABC$ given the length of $AB, BC$ and the size of $\angle A$ , if this triangle has one or more of the conditions below, then $SSA$ is true:

$\dfrac { BC }{ AB }=\sin { \angle A }$
$\angle A \ge 90\degree$
$AB\ge BC$

I hope it will help you guys! Please comment below if you had something to say about this article. Thank you!

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Rocky Writer - 4 days, 20 hours ago

Yes definitely, it is very helpful for the student because mostly students face problems in SSA equations. They can easy to solve it but when the question is about 2 triangles so they never prove it. This equation is very hectic and technical that is required lots of skills and strategy. I have good experience in this question because I also prepared students' homework because I am a writer associated with cheap assignment writing uk so this is why I have much knowledge about these questions. By the way, it is an excellent work and I hope students will take the lesson.

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Wooten Huber - 3 weeks, 5 days ago

You might be tempted to think that given two sides and a non-included angle is enough to prove congruence. But there are two triangles possible that have the same values, so SSA is not sufficient to prove congruence. I am a content writer if you want uk assignment writing service contact me I am here to assist you.

James Tinker - 3 weeks, 5 days ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$