Is 70°80° 30° a unique triangle, many triangle or no triangle

Question

Bradyn Mills

Mathematics

17 May, 18:45

Is 70°80° 30° a unique triangle, many triangle or no triangle

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Kimberly Watts · Answer 1

Many triangles.

Step-by-step explanation:

The fact that the three angles add up to 180 degrees allows you to say this is a valid triangle proposal.

But because we are not given a size of any of the three sides, you can reject the proposal that this is a single triangle, and it is easy to show why: Suppose there is a single triangle with these angles. Then double each of its three sides. Now we have two triangles with the same three angles. According to our assumption, they must be congruent! Well, they are similar, for starters. However, none of our known congruence postulates (ASA, SAS) apply because the two triangles have every side of different length! This implies they cannot be guaranteed to be congruent which is a contradiction and the the two triangles (original and doubled one) are not the same, yet they have the same angles. Therefore, we can construct many (actually, infinite number of) triangles with these angles.