Question

for each of the following conditions, is it possible to construct two noncongruent triangles? explain why or why not. a. a triangle with angles measuring 60° and 70° and an included side of 8 in. b. a triangle with angles measuring 60° and 70° and a non - included side of 8 cm on a side of the 60° angle c. a triangle with angles measuring 30°, 70°, and 80°. a. choose the correct answer below. it is not possible, because any two triangles with these conditions are congruent by aas. it is not possible, because any two triangles with these conditions are congruent by sas. it is not possible, because any two triangles with these conditions are congruent by asa. it is possible, because the given conditions determine a unique shape but not size. b. choose the correct answer below. a. it is not possible, because any two triangles with these conditions are congruent by aas. b. it is not possible, because any two triangles with these conditions are congruent by sas. c. it is not possible, because any two triangles with these conditions are congruent by asa. d. it is possible, because the given conditions determine a unique shape but not size.

Explanation:

Step1: Recall triangle - congruence postulates

The ASA (Angle - Side - Angle) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. In part a, we have two angles ($60^{\circ}$ and $70^{\circ}$) and their included side (8 in), so by ASA, any two triangles with these conditions are congruent.

Step2: Analyze part b

For part b, when we have two angles ($60^{\circ}$ and $70^{\circ}$) and a non - included side of 8 cm on a side of the $60^{\circ}$ angle, we can use the Law of Sines. But in this case, the given information still determines a unique triangle. The AAS (Angle - Angle - Side) postulate can be applied. Once we know two angles, the third angle is fixed ($180-(60 + 70)=50^{\circ}$), and with a non - included side given, the triangle is unique.

Step3: Analyze part c

For part c, when only the angles ($30^{\circ}$, $70^{\circ}$, and $80^{\circ}$) are given, we can construct infinitely many non - congruent similar triangles. Since we have no side - length information, the triangles can have different sizes but the same shape.

Answer:

a. It is not possible, because any two triangles with these conditions are congruent by ASA
b. It is not possible, because any two triangles with these conditions are congruent by AAS
c. It is possible, because only angle measures are given and no side - length information, so non - congruent similar triangles can be constructed.