Question

Which dimensions can create only one unique triangle?

Answer 1

Answer:

B)

Step-by-step explanation:

A, C, and D multiple choice answers have no chance of becoming a triangle because the lengths don't add up to either 180 or 90. Also, Triangle B follows the rule that 2 sides and be greater than the other side.

Answer 2

Answer: Choice B)

3 sides measuring 14 m, 11 m, and 18 m

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Explanation:

Let's go through choices A through D to see which result in one unique triangle, or could produce (infinitely) many triangles, or produce no triangles at all.

• Choice A produces no triangles at all. Why is this? The angles don't add up to 180. We see that 75+45+50 = 170. The three angles must add to 180 degrees for a triangle to form.
• Choice B produces one single triangle, which is why it's the final answer. The triangle inequality theorem can be used here to show that a triangle can be formed. The SSS (side side side) congruence theorem is then used to show that exactly one unique triangle is possible.
• Choice C is a similar story to choice A. We can rule it out. This time we have the angles add to 40+50+60 = 150 which is short of 180.
• Choice D can be ruled out too. The two sides a = 3 and b = 4 add to a+b = 3+4 = 7, but that's smaller than c = 8. We must have a+b > c to be true if we wanted to form a triangle. For more info, see the triangle inequality theorem.

Even if the angles did add to 180 (for either choice A or choice C), then we'd have infinitely many triangles and not one unique triangle. Think of dilating the triangles and how each dilated image retains the same angles. In other words, the triangles keep the same shape. Simply knowing the three angles is not enough info to create one unique triangle. We would need to know some info about the sides. This explains why AA or AAA is not a congruence theorem.