Question

δxyz was reflected to form δlmn. which statements are true regarding the diagram? check all that apply. □ δxyz ≅ δlmn □ ∠y ≅ ∠m □ ∠x ≅ ∠l □ ∠z ≅ ∠l □ \overline{yz} ≅ \overline{ml} □ \overline{xz} ≅ \overline{ln}

Explanation:

Step1: Recall Reflection Property

Reflection is a rigid transformation, so reflected figures are congruent ($\triangle XYZ \cong \triangle LMN$). Corresponding angles and sides are congruent.

Step2: Find Angles in $\triangle XYZ$

In $\triangle XYZ$, $\angle Y = 86^\circ$, $\angle X = 38^\circ$, so $\angle Z = 180 - 86 - 38 = 56^\circ$.
In $\triangle LMN$, $\angle M = 86^\circ$, $\angle N = 56^\circ$, so $\angle L = 180 - 86 - 56 = 38^\circ$.

Step3: Analyze Each Statement

• $\triangle XYZ \cong \triangle LMN$: True (reflection preserves congruence).
• $\angle Y \cong \angle M$: $\angle Y = 86^\circ$, $\angle M = 86^\circ$ → True.
• $\angle X \cong \angle L$: $\angle X = 38^\circ$, $\angle L = 38^\circ$ → True.
• $\angle Z \cong \angle L$: $\angle Z = 56^\circ$, $\angle L = 38^\circ$ → False.
• $\overline{YZ} \cong \overline{ML}$: Corresponding sides? $YZ$ corresponds to $MN$ (since $\angle Y$ corresponds to $\angle M$, $\angle Z$ to $\angle N$, $\angle X$ to $\angle L$), so $YZ$ and $ML$ not corresponding → False.
• $\overline{XZ} \cong \overline{LN}$: $XZ$ (between $\angle X$ and $\angle Z$) corresponds to $LN$ (between $\angle L$ and $\angle N$) → $\angle X = \angle L = 38^\circ$, $\angle Z = \angle N = 56^\circ$, so sides correspond → True.

Answer:

$\triangle XYZ \cong \triangle LMN$, $\angle Y \cong \angle M$, $\angle X \cong \angle L$, $\overline{XZ} \cong \overline{LN}$ (i.e., the first, second, third, and sixth options)