Question

decide if each pair of triangles below is similar. if the triangles are similar, justify your conclusion by stating the similarity condition you used. also describe a possible sequence of transformations that would carry one onto the other. if the triangles are not similar, explain how you know.
hint (a):
hint (b):
equilateral triangles
hint (c):
hint (d):

Explanation:

Step1: Check for AA (Angle - Angle) similarity in part (a)

In the two right - angled triangles, one of the non - right angles (\(x\)) is common. Also, both have a right angle (\(90^{\circ}\)). By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Step2: Check for similarity in part (b)

All equilateral triangles have all angles equal to \(60^{\circ}\). Let the first equilateral triangle have side lengths \(a,b,c\) and the second have side lengths \(A,B,C\). The ratio of corresponding sides \(\frac{a}{A}=\frac{b}{B}=\frac{c}{C}\) (since all sides of an equilateral triangle are equal within the triangle, and for two equilateral triangles, if we assume side lengths \(s_1\) and \(s_2\), the ratio of sides is a constant). By the SSS (Side - Side - Side) similarity criterion (if the ratios of the corresponding sides of two triangles are equal, the triangles are similar) or AA (since all angles are \(60^{\circ}\)), the equilateral triangles are similar.

Step3: Check for similarity in part (c)

Calculate the ratios of the corresponding sides. \(\frac{6}{15}=\frac{2}{5}\), \(\frac{8}{20}=\frac{2}{5}\), \(\frac{11}{33}=\frac{1}{3}\). Since \(\frac{6}{15}=\frac{8}{20}
eq\frac{11}{33}\), the ratios of the corresponding sides are not equal. So, the triangles are not similar.

Step4: Check for AA similarity in part (d)

For the first triangle, the third angle is \(180-(60 + 50)=70^{\circ}\). For the second triangle, the third angle is \(180-(30 + 25)=125^{\circ}\). There are no two pairs of equal angles. So, the triangles are not similar.

Answer:

• a. The triangles are similar by the AA (Angle - Angle) similarity criterion.
• b. The equilateral triangles are similar. They satisfy the SSS (Side - Side - Side) similarity criterion (ratios of all corresponding sides are equal as all sides of an equilateral triangle are in proportion) or AA (all angles are \(60^{\circ}\)).
• c. The triangles are not similar because the ratios of the corresponding sides \(\frac{6}{15}=\frac{8}{20}

eq\frac{11}{33}\) are not all equal.

• d. The triangles are not similar because there are no two pairs of equal angles.