Question

triangle abc is shown on the coordinate plane. triangle abc is reflected over the x - axis and then translated vertically to create triangle def. which statements must be true? select the two correct statements. a ∠f ≅ ∠e b ∠e ≅ ∠b c ∠d ≅ ∠c d m∠c = m∠e e m∠b = m∠c f m∠a = m∠d

Explanation:

Step1: Properties of Reflection and Translation

Reflection over the \(x -\)axis and translation (both are rigid motions) preserve the shape and size of the figure. So, \(\triangle ABC\cong\triangle DEF\).
When two triangles are congruent, their corresponding angles are congruent. That is, \(\angle A\cong\angle D\), \(\angle B\cong\angle E\), \(\angle C\cong\angle F\).
Congruent angles have equal measures. So, \(m\angle A = m\angle D\), \(m\angle B=m\angle E\), \(m\angle C = m\angle F\).

Step2: Analyzing Each Option

• Option A: There is no information to suggest \(\angle F\cong\angle E\) from the congruence of \(\triangle ABC\) and \(\triangle DEF\).
• Option B: Since \(\triangle ABC\cong\triangle DEF\), \(\angle E\cong\angle B\) (corresponding angles of congruent triangles).
• Option C: \(\angle D\cong\angle A\) (corresponding angles of congruent triangles), not \(\angle D\cong\angle C\).
• Option D: \(m\angle C=m\angle F

eq m\angle E\) (from the congruence \(\triangle ABC\cong\triangle DEF\)).

• Option E: There is no information to suggest \(m\angle B = m\angle C\) from the congruence of \(\triangle ABC\) and \(\triangle DEF\).
• Option F: Since \(\triangle ABC\cong\triangle DEF\), \(m\angle A=m\angle D\) (corresponding angles of congruent triangles).

Answer:

B. \(\angle E\cong\angle B\), F. \(m\angle A = m\angle D\)