Question

1. △abc is dilated from center a by a factor not equal to 1 to form △akl. which of the statements must be true? select all that apply. a. \\(\overline{ab}\\) and \\(\overline{ak}\\) lie on the same line. b. the line containing \\(\overline{bc}\\) is parallel to the line containing \\(\overline{kl}\\). c. \\(\angle abc \cong \angle acb\\) d. \\(\angle abc \cong \angle akl\\) e. \\(\triangle abc \sim \triangle akl\\) f. \\(\triangle abc \cong \triangle akl\\)

Explanation:

Step1: Properties of Dilation

Dilation is a transformation that maps a line segment to a parallel line segment (if the scale factor is not \(1\)). Also, points on the same ray from the center of dilation lie on the same line.

• For option A: By the definition of dilation, \(\overline{AB}\) and \(\overline{AK}\) are on the same ray from center \(A\), so they lie on the same line.
• For option B: Since \(\triangle ABC\) is dilated to \(\triangle AKL\), by the property of dilation, \(\overline{BC}\parallel\overline{KL}\) (because dilation preserves parallelism when the scale factor \(k

eq1\)).

• For option D: \(\angle ABC\) and \(\angle AKL\) are corresponding angles. Dilation is a similarity transformation. Similarity transformations preserve angle - measures. So \(\angle ABC\cong\angle AKL\).
• For option E: Dilation is a similarity transformation. So \(\triangle ABC\sim\triangle AKL\) (by the AA similarity criterion, since \(\angle A=\angle A\) (common angle) and \(\angle ABC\cong\angle AKL\) as shown above).

Step2: Analyzing incorrect options

• Option C: \(\angle ABC\) and \(\angle ACB\) are angles in \(\triangle ABC\). There is no reason for \(\angle ABC\cong\angle ACB\) based on the dilation transformation (unless \(\triangle ABC\) is isosceles, which is not given).
• Option F: Dilation is a similarity transformation. If the scale factor \(k = 1\), \(\triangle ABC\cong\triangle AKL\), but since the scale factor is not equal to \(1\) (given in the problem statement), \(\triangle ABC

ot\cong\triangle AKL\).

Answer:

A. \(\overline{AB}\) and \(\overline{AK}\) lie on the same line.
B. The line containing \(\overline{BC}\) is parallel to the line containing \(\overline{KL}\).
D. \(\angle ABC\cong\angle AKL\)
E. \(\triangle ABC\sim\triangle AKL\)