Question

given triangle abc and triangle def, with ∠a = 60 degrees and ∠d = 60 degrees, if ab/de = 3 and ac/df = 3, are the triangles similar? a. no, because only one - angle is given b. no, because the sides are not proportional c. yes, by aaa similarity d. yes, by sas similarity when applying sas similarity, which of the following must be true? a. one angle and one side must be equal b. all angles must be congruent c. two sides must be equal d. the sides around the congruent angles must be proportional if two triangles have one pair of congruent angles and the surrounding sides are in a ratio of 4, are the triangles similar? a. yes, by sas similarity b. yes, by sss similarity c. no, they must have two pairs of congruent angles d. no, they must have three pairs of congruent angles

Explanation:

Step1: Recall similarity - criteria

The Side - Angle - Side (SAS) similarity criterion states that if two sides of one triangle are proportional to two corresponding sides of another triangle and the included angles are congruent, then the two triangles are similar.

Step2: Analyze first question

In triangles ABC and DEF, $\angle A=\angle D = 60^{\circ}$, and $\frac{AB}{DE}=3$, $\frac{AC}{DF}=3$. The sides AB and AC are around $\angle A$, and DE and DF are around $\angle D$. Since the included angles are congruent and the sides around them are proportional, by SAS similarity, the triangles are similar.

Step3: Analyze second question

For SAS similarity, the sides around the congruent angles must be proportional.

Step4: Analyze third question

If two triangles have one pair of congruent angles and the surrounding sides are in a ratio (in this case, the ratio is 4), by the SAS similarity criterion, the triangles are similar.

Answer:

1. d. Yes, by SAS similarity
2. d. The sides around the congruent angles must be proportional
3. a. Yes, by SAS similarity