Question

determine whether the following pairs of triangles, (i)-(iv), are similar. if they are similar, explain why. find the scale factor of the triangles that are similar. the figures are not drawn to scale.
(i) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
oa. △abc∼△def by the sss similarity for triangles theorem. the scale factor of △abc to △def is
(type an integer or a simplified fraction.)
ob. △abc∼△def by the aa similarity for triangles theorem. the scale factor of △abc to △def is
(type an integer or a simplified fraction.)
oc. △abc∼△def by the sas similarity for triangles theorem. the scale factor of △abc to △def is
(type an integer or a simplified fraction.)
od. the triangles are not similar

Explanation:

Step1: Find the third angle of \(\triangle ABC\)

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle of \(\triangle ABC\) be \(x\). Then \(x + 31^{\circ}+62^{\circ}=180^{\circ}\), so \(x=180-(31 + 62)=87^{\circ}\).

Step2: Find the third angle of \(\triangle DEF\)

Let the third angle of \(\triangle DEF\) be \(y\). Then \(y+31^{\circ}+62^{\circ}=180^{\circ}\), so \(y = 180-(31 + 62)=87^{\circ}\).
Since two angles of \(\triangle ABC\) (\(31^{\circ}\) and \(62^{\circ}\)) are equal to two angles of \(\triangle DEF\) (\(31^{\circ}\) and \(62^{\circ}\)), by the AA (Angle - Angle) similarity theorem, \(\triangle ABC\sim\triangle DEF\).

Step3: Calculate the scale factor

The scale factor is the ratio of corresponding sides. Let's assume the side of length \(1\) in \(\triangle ABC\) corresponds to the side of length \(6\) in \(\triangle DEF\). The scale factor of \(\triangle ABC\) to \(\triangle DEF\) is \(\frac{1}{6}\)

Answer:

B. \(\triangle ABC\sim\triangle DEF\) by the AA Similarity for Triangles theorem. The scale factor of \(\triangle ABC\) to \(\triangle DEF\) is \(\frac{1}{6}\)