Question

which rule explains why these scalene triangles are similar?
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Check the ratio of sides

Calculate the ratios of corresponding sides. For \(\triangle PQR\) and \(\triangle VWU\), \(\frac{PQ}{WU}=\frac{63}{30}=\frac{21}{10}\), \(\frac{PR}{WV}=\frac{42}{45}=\frac{14}{15}\). The ratios of sides are not equal. So SSS (Side - Side - Side) similarity is not applicable.

Step2: Check SAS similarity

SAS similarity requires the ratio of two sides and the included angle. We have \(\frac{PQ}{WU}=\frac{63}{30}=\frac{21}{10}\), \(\frac{PR}{WV}=\frac{42}{45}=\frac{14}{15}\). The ratios of the two pairs of sides are not equal. So SAS (Side - Angle - Side) similarity is not applicable.

Step3: Check for AA similarity

We know that in a triangle, the sum of angles is \(180^{\circ}\). But we are given one angle (\(96^{\circ}\)) in each triangle. Since we have only one angle given and no information about the other angles (to check for AA - Angle - Angle similarity where two angles of one triangle are equal to two angles of another triangle), we cannot prove the triangles are similar using AA.

Answer:

none of the above; the triangles cannot be proven similar