Question

3 - 54 decide if each pair of triangles below is similar. if the triangles are similar, justify your conclusion by stating the similarity condition you used (aa or sas). also describe a possible sequence of transformations that would carry one onto the other (reflection, rotation, translation, dilation). if the triangles are not similar, explain how you know.
equilateral triangles
a.
b.
c.
d.

Explanation:

Step1: Analyze pair a

• AA similarity criterion:
• In the two right - angled triangles (pair a), we have two right angles (equal). Also, the common non - right angle (by the property of angles in a triangle, if two angles of one triangle are equal to two angles of another triangle, the third angles are also equal).
• So, by AA (Angle - Angle) similarity criterion, the triangles are similar. A possible transformation is a dilation (scaling) as the triangles are of different sizes but same shape.

Step2: Analyze pair b

• Definition of equilateral triangles:
• All equilateral triangles have all angles equal to \(60^{\circ}\).
• Let the side lengths of the larger equilateral triangle be \(s_1\) and of the smaller one be \(s_2\). The ratio of their corresponding sides is constant (\(\frac{s_1}{s_2}\)).
• By AA similarity (since all angles are equal), the triangles are similar. A possible transformation is a dilation.

Step3: Analyze pair c

• Check side - side - side ratios:
• Calculate the ratios of the corresponding sides: \(\frac{9}{15}=\frac{3}{5}\), \(\frac{11}{25}

eq\frac{3}{5}\), \(\frac{9}{15}
eq\frac{11}{25}\).

• Since the ratios of the corresponding sides are not equal, the triangles are not similar.

Step4: Analyze pair d

• Check angle - side - angle:
• One triangle has angles \(60^{\circ},60^{\circ},60^{\circ}\) (equilateral, all angles \(60^{\circ}\)) and the other has angles \(60^{\circ},60^{\circ},60^{\circ}\) (equilateral). But the side lengths: \(\frac{4}{5}

eq1\) (if we assume side lengths 4 and 5 for non - matching sides).

• Wait, no. Wait, for the two triangles in pair d:
• Let's use the angle - angle similarity. But also check side ratios.
• The first triangle has angles \(60^{\circ},60^{\circ},60^{\circ}\) (equilateral) and the second has angles \(60^{\circ},60^{\circ},60^{\circ}\) (equilateral). But the side lengths: \(\frac{4}{5}

eq1\). Wait, no, actually, if we consider the angles:

• All angles are equal (\(60^{\circ}\) each). So by AA similarity, they are similar. A possible transformation is a dilation.

Answer:

• a. Similar (by AA similarity, transformation: dilation).
• b. Similar (by AA similarity, transformation: dilation).
• c. Not similar (side - side - side ratios not equal).
• d. Similar (by AA similarity, transformation: dilation).