Question

1. reflect right triangle abc across line bc. classify triangle aca according to its side lengths. explain how you know.
2. 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: Analyze option A

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

Step2: Analyze option B

\(\angle BCA\) and \(\angle CED\): Since \(DCE\) is a translation of \(ABC\), translation preserves angle measures. \(\angle BCA=\angle CED\)

Step3: Analyze option C

\(AD\) is not parallel to \(BC\). If we consider the translation vectors, we can see that the direction of \(AD\) is not the same as \(BC\)

Step4: Analyze option D

\(\angle CED\) (from \(\triangle DCE\)) and \(\angle FAD\) (from \(\triangle FAD\)). Since \(\triangle FAD\) and \(\triangle DCE\) are translations of \(\triangle ABC\), and translation preserves angle measures. \(\angle CED=\angle FAD\)

Step5: Analyze option E

\(\angle DAC\) and \(\angle BCA\): There is no translation or congruence relation that would make these angles equal.

Step6: Analyze option F

\(\triangle ADC\) and \(\triangle FAD\): There is no reflection transformation that maps \(\triangle FAD\) to \(\triangle ADC\)

Answer:

A. Points \(B\), \(A\), and \(F\) are collinear.
B. The measure of angle \(BCA\) is the same as the measure of angle \(CED\).
D. The measure of angle \(CED\) is the same as the measure of angle \(FAD\).