Question

3 reflect right triangle abc across line bc. classify triangle aca according to its side lengths. explain how you know. 4 from unit 1, lesson 21 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 reflection property

When right - triangle $ABC$ is reflected across line $BC$ to get $A'$, we know that $CA = CA'$ because reflection is a rigid transformation that preserves distances.

Step2: Classify triangle $ACA'$

Since $CA = CA'$, triangle $ACA'$ has two equal side - lengths. By the definition of an isosceles triangle (a triangle with at least two equal side - lengths), triangle $ACA'$ is isosceles.

For question 4:

Step1: Recall properties of translations

Translations are rigid transformations that preserve angle measures and parallel relationships.

• Since $\triangle FAD$ and $\triangle DCE$ are translations of $\triangle ABC$:
• For option A: There is no information to suggest that points $B$, $A$, and $F$ are collinear. Translations do not necessarily place these points on the same line.
• For option B: Since $\triangle DCE$ is a translation of $\triangle ABC$, corresponding angles are equal. $\angle BCA$ and $\angle CED$ are corresponding angles, so the measure of angle $BCA$ is the same as the measure of angle $CED$.
• For option C: In translations, corresponding sides are parallel. In the translation from $\triangle ABC$ to $\triangle FAD$ and $\triangle DCE$, line $AD$ is parallel to line $BC$.
• For option D: $\angle CED$ and $\angle FAD$ are not corresponding angles in the translations, so we cannot say they are equal.
• For option E: $\angle DAC$ and $\angle BCA$ are alternate - interior angles for parallel lines $AD$ and $BC$ (from the translation property), so the measure of angle $DAC$ is the same as the measure of angle $BCA$.
• For option F: There is no indication that $\triangle ADC$ is a reflection of $\triangle FAD$. Translations and reflections are different transformations, and the given information about translations does not imply a reflection relationship.

Answer:

1. Triangle $ACA'$ is isosceles because when $\triangle ABC$ is reflected across line $BC$, $CA = CA'$ by the property of reflection.
2. B. The measure of angle $BCA$ is the same as the measure of angle $CED$.

C. Line $AD$ is parallel to line $BC$.
E. The measure of angle $DAC$ is the same as the measure of angle $BCA$.