Question

determine whether the following pairs of triangles, (i)-(vi), 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) the triangles are not similar.
(iii) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. ( \triangle acdsim\triangle abe ) by the aa similarity for triangles theorem. the scale factor of ( \triangle acd ) to ( \triangle abe ) is (type an integer or a simplified fraction.)
b. ( \triangle acdsim\triangle abe ) by the sss similarity for triangles theorem. the scale factor of ( \triangle acd ) to ( \triangle abe ) is (type an integer or a simplified fraction.)
c. ( \triangle acdsim\triangle abe ) by the sas similarity for triangles theorem. the scale factor of ( \triangle acd ) to ( \triangle abe ) is (type an integer or a simplified fraction.)
d. the triangles are not similar

Explanation:

Step1: Identify common angle

Both \(\triangle ACD\) and \(\triangle ABE\) share \(\angle A\).

Step2: Check for another equal angle

\(\angle ABE=\angle ACD\) (given as marked in the figure).

Step3: Apply AA similarity theorem

Since two angles of \(\triangle ACD\) are equal to two angles of \(\triangle ABE\), by AA (Angle - Angle) similarity theorem, \(\triangle ACD\sim\triangle ABE\).

Step4: Calculate scale factor

\(AC = 4 + 1=5\), \(AB = 4\). The scale factor of \(\triangle ACD\) to \(\triangle ABE\) is \(\frac{AC}{AB}=\frac{5}{4}\).

Answer:

A. \(\triangle ACD\sim\triangle ABE\) by the AA Similarity for Triangles theorem. The scale factor of \(\triangle ACD\) to \(\triangle ABE\) is \(\frac{5}{4}\)