Question

How many different triangles can you make with the following conditions: angles measuring 45 degrees and 90 degrees and an INCLUDED side that is 1 inch? one more than one none

Answer 1

Answer:

One

Step-by-step explanation:

Clearly, one triangle can be constructed as the angles 45 and 90 do not exceed 180 degrees. (so "None" is not correct)

To show that only one such triangle exists, you can apply the Angle-Side-Angle theorem for congruence.

Since one triangle can be constructed, it remains to be shown that no additional triangle that is not congruent to the first one can be created: I will use proof by contradiction. Let a triangle ABC be constructed with two angles 45 and 90 degree and one included side of length 1 inch. Suppose, I now construct a second triangle that is different from the first one but still has the same two angles and included side. By applying the ASA theorem which states that two triangles with same two angles and one side included are congruent, I must conclude that my triangle is congruent to the first one. This is a contradiction, hence my original claim could not have been true. Therefore, there is no way to construct any additional triangle that would not be congruent with the first one, and only one such triangle exists.