Question

which relationships within the diagram are true? check all that apply. □ fc≅ac □ fc≅be □ ∠cfa≅∠eba □ △acf≅△aeb because of aas. □ △acf≅△aeb because of sas. □ △acf≅△aeb because of asa. ac≅ae; ∠acd≅∠aed

Explanation:

Step1: Recall triangle - congruence postulates

AAS (Angle - Angle - Side) means two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle. SAS (Side - Angle - Side) means two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle. ASA (Angle - Side - Angle) means two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

Step2: Analyze each option

• $\overline{FC}\cong\overline{AC}$: There is no information in the given congruences ($\overline{AC}\cong\overline{AE}$, $\angle ACD\cong\angle AED$) to suggest that $\overline{FC}\cong\overline{AC}$, so this is false.
• $\overline{FC}\cong\overline{BE}$: There is no information in the given congruences to suggest this, so this is false.
• $\angle CFA\cong\angle EBA$: There is no information in the given congruences to suggest this, so this is false.
• $\triangle ACF\cong\triangle AEB$ because of AAS: We know $\overline{AC}\cong\overline{AE}$ and $\angle ACD\cong\angle AED$. Also, $\angle A$ is common to both $\triangle ACF$ and $\triangle AEB$. So, by AAS (two angles and a non - included side), $\triangle ACF\cong\triangle AEB$. This is true.
• $\triangle ACF\cong\triangle AEB$ because of SAS: We do not have two sides and the included angle congruent based on the given $\overline{AC}\cong\overline{AE}$ and $\angle ACD\cong\angle AED$, so this is false.
• $\triangle ACF\cong\triangle AEB$ because of ASA: We do not have two angles and the included side congruent based on the given $\overline{AC}\cong\overline{AE}$ and $\angle ACD\cong\angle AED$, so this is false.

Answer:

$\triangle ACF\cong\triangle AEB$ because of AAS