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 ΔABC such that we only know about the length of AB,BC and the size of ∠A, by the sine formula, we can show that
sin∠ABC=sin∠CAB→sin∠C=BCABsin∠A
From here there are two cases:
Case 1, sin∠C=1
There are only 1 possibility of ∠C, which is 90°. Then, we know the size of ∠A,∠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∠C=BCABsin∠A=1, which means ABBC=sin∠A, SSA is true.
Case 2, 0<sin∠C<1
Then, there are 2 possibilities of ∠C, sin−1(BCABsin∠A) and 180°−sin−1(BCABsin∠A). Hence, we need to consider ∠A to find out more conditions of proving SSA. By the angle sum of triangle, ∠C=180°−∠A−∠B<180°−∠A. We will consider two cases about ∠A
Case A, ∠A≥90°
∠C<180°−∠A≤90°. Then, ∠C can only be sin−1(BCABsin∠A), so there are only 1 possible triangle similarly. Therefore, we can conclude that when ∠A≥90°, SSA is true.
Do you notice that? When ∠A=90°, it is the same case as RHS (Right angle-hypotenuse-side) or HL (Hypotenuse-leg)
Case B, ∠A<90°
∠C<180°−∠A. It is true when ∠C=sin−1(BCABsin∠A). Because of we want to have 1 possibility of ∠C, we need to ignore the other possibility of ∠C, which we get the inequality below:
90°>180°−sin−1(BCABsin∠A)≥180°−∠Asin90°>sin(180°−sin−1(BCABsin∠A))≥sin(180°−∠A)BCABsin∠A≥sin∠AAB≥BC
Therefore, we can conclude that when AB≥BC, SSA is true.
To sum up, there is a triangle ΔABC given the length of AB,BC and the size of ∠A, if this triangle has one or more of the conditions below, then SSA is true:
- ABBC=sin∠A
- ∠A≥90°
- AB≥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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Top NewestHow 90>180−sin−1(acsinA)≥180−A is true ?
As, A<90 So 180−A>90
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A great blog you provided for students it will be very helpful for their study.so, keep posting. https://assignmenthelps.co.uk/
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I didn't provide any blog...
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There are many cases it doesn't prove congruency.
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The SSA or ASS combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem. [url=https://royalbritishessaywriters.co.uk/]business essay writing service[/url] SSA or ASS is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same.
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