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.
(from unit 1, lesson 21.)

Explanation:

Step1: Recall properties of translation

Translation is a rigid transformation. So, \(\triangle FAD\cong\triangle ABC\cong\triangle DCE\). Corresponding angles and sides are equal.

Step2: Analyze option A

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

Step3: Analyze option B

\(\angle BCA\) and \(\angle CED\) are not corresponding angles of congruent triangles. So, this is not necessarily true.

Step4: Analyze option C

Since \(\triangle FAD\cong\triangle ABC\cong\triangle DCE\), \(AD = BC\) and \(AD\parallel BC\) (because translation preserves parallelism).

Step5: Analyze option D

\(\angle CED\) and \(\angle FAD\) are not corresponding angles of congruent triangles. So, this is not necessarily true.

Step6: Analyze option E

\(\angle DAC\) and \(\angle BCA\) are alternate - interior angles (because \(AD\parallel BC\) and \(AC\) is a transversal). So, \(\angle DAC=\angle BCA\).

Step7: Analyze option F

There is no information to suggest that \(\triangle ADC\) is a reflection of \(\triangle FAD\). Reflection is a different transformation from translation.

Answer:

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