Question

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: Recall translation property

Translations preserve angle - measures and parallelism. Triangles FAD and DCE are translations of triangle ABC.

Step2: Analyze option A

Since FAD is a translation of ABC, and translation is a rigid - motion that moves points in a straight line, points B, A, and F are collinear.

Step3: Analyze option B

Because DCE is a translation of ABC, corresponding angles are equal. ∠BCA and ∠CED are corresponding angles, so m∠BCA=m∠CED.

Step4: Analyze option C

In a translation, corresponding line - segments are parallel. AD is a corresponding line - segment to BC in the translation, so AD∥BC.

Step5: Analyze option D

∠CED corresponds to ∠BCA in the translation of ABC to DCE, and ∠FAD corresponds to ∠BAC in the translation of ABC to FAD. There is no reason for ∠CED and ∠FAD to be equal.

Step6: Analyze option E

There is no relationship based on translation that would make ∠DAC and ∠BCA equal.

Step7: Analyze option F

There is no indication that triangle ADC is a reflection of triangle FAD. Translations and reflections are different rigid - motions, and just because FAD is a translation of ABC does not mean ADC is a reflection of FAD.

Answer:

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.