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°. 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. c. choose the correct answer below. a. it is not possible, because any two triangles with these conditions are congruent by sas. b. it is not possible, because any two triangles with these conditions are congruent by aas. 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 congruence postulates

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, the triangles are congruent. AAS (Angle - Angle - Side) postulate states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent.

Step2: Analyze part b

In part b, 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. By the AAS congruence postulate, any two triangles with these conditions are congruent. So it is not possible to construct two non - congruent triangles.

Step3: Analyze part c

In part c, we are given only the three angle measures ($30^{\circ}$, $70^{\circ}$, and $80^{\circ}$). The angle - angle (AA) similarity criterion is used here. Since we have no information about the side lengths, we can have infinitely many non - congruent triangles with the same angle measures (similar triangles). The given conditions determine a unique shape (because the angles are fixed) but not size. So it is possible to construct non - congruent triangles.

Answer:

b. It is not possible, because any two triangles with these conditions are congruent by AAS
c. It is possible, because the given conditions determine a unique shape but not size