Question

1. triangles fad and dce are translations of triangle abc. select all the statements that must be true. a. points b, a, and f are collinear. b. the measure of angle bca is the same as the measure of angle ced. c. line ad is parallel to line bc. d. the measure of angle ced is the same as the measure of angle fad. e. the measure of angle dac is the same as the measure of angle bca. f. triangle adc is a reflection of triangle fad.

Explanation:

Step1: Analyze option A

Since \(FAD\) is a translation of \(ABC\), \(BA\) and \(FA\) are part of the same line. So, points \(B\), \(A\), and \(F\) are collinear.

Step2: Analyze option B

\(\angle BCA\) and \(\angle CED\) are not corresponding angles in the translation. Translations preserve angle measures for corresponding angles. \(\angle BCA\) corresponds to \(\angle DAC\) (from \(ABC\) to \(DAC\) translation - if we consider the translation concept).

Step3: Analyze option C

Since \(FAD\) and \(DCE\) are translations of \(ABC\), \(AD\parallel BC\) (translation preserves parallelism. If \(ABC\) is translated to \(FAD\) and \(DCE\), the side - side relationships for parallelism are maintained).

Step4: Analyze option D

\(\angle CED\) and \(\angle FAD\) are not corresponding angles. \(\angle FAD\) corresponds to \(\angle ABC\) (from translation of \(ABC\) to \(FAD\)) and \(\angle CED\) corresponds to \(\angle ABC\) (from translation of \(ABC\) to \(DCE\)). So, \(\angle CED=\angle FAD\) (transitive property as both are equal to \(\angle ABC\)).

Step5: Analyze option E

Since \(ABC\) is translated to \(DAC\) (assuming the translation vector), \(\angle DAC=\angle BCA\) (translation preserves angle measures).

Step6: Analyze option F

If we consider the translation of \(\triangle FAD\) to \(\triangle ADC\), a translation (not a reflection) would map \(\triangle FAD\) to \(\triangle ADC\) as the orientation (in terms of side - side and angle - angle relationships without a flip) is maintained.

Answer:

A. Points \(B\), \(A\), and \(F\) are collinear.
C. Line \(AD\) is parallel to line \(BC\).
D. The measure of angle \(CED\) is the same as the measure of angle \(FAD\).
E. The measure of angle \(DAC\) is the same as the measure of angle \(BCA\).