Question

practice using asa and aas congruence theorems.
what additional information could be used to prove that
△xyz ≅ △feg using asa or aas? check all that apply.
□ ∠z ≅ ∠g and overline{xz} ≅ overline{fg}
□ ∠z ≅ ∠g and ∠y ≅ ∠e
□ overline{xz} ≅ overline{fg} and overline{zy} ≅ overline{ge}
□ overline{xy} ≅ overline{ef} and overline{zy} ≅ overline{fg}
□ ∠z ≅ ∠g and overline{xy} ≅ overline{fe}

Explanation:

Step1: Recall ASA and AAS congruence criteria

ASA (Angle - Side - Angle) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. AAS (Angle - Angle - Side) states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.

Step2: Analyze each option

• Option 1: \(\angle Z\cong\angle G\) and \(XZ\cong FG\)

If \(\angle Z\cong\angle G\) and \(XZ\cong FG\), and we assume the angle at \(X\) is congruent to the angle at \(F\) (from the given markings in the figure), then by ASA (if the side is between two angles) or AAS (if we consider the non - included side case), this information can be used.

• Option 2: \(\angle Z\cong\angle G\) and \(ZY\cong GE\)

If \(\angle Z\cong\angle G\) and \(ZY\cong GE\), and \(\angle X\cong\angle F\) (from the figure), then by AAS (two angles and a non - included side), this information can be used.

• Option 3: \(\overline{XZ}\cong\overline{FG}\) and \(\overline{ZY}\cong\overline{GE}\)

This is a Side - Side (two sides) situation. ASA and AAS require at least two angles. So, this option does not work for ASA or AAS.

• Option 4: \(\overline{XY}\cong\overline{EF}\) and \(\overline{ZY}\cong\overline{FG}\)

This is also a Side - Side (two sides) situation. ASA and AAS require at least two angles. So, this option does not work for ASA or AAS.

• Option 5: \(\angle Z\cong\angle G\) and \(\overline{XY}\cong\overline{FE}\)

This is an Angle - Side (one angle and one side) situation. ASA and AAS require at least two angles. So, this option does not work for ASA or AAS.

Answer:

\(\angle Z\cong\angle G\) and \(XZ\cong FG\), \(\angle Z\cong\angle G\) and \(ZY\cong GE\)