Question

the proof that △mng ≅ △kjg is shown. given: ∠n and ∠j are right angles; ng ≅ jg prove: △mng ≅ △kjg what is the missing reason in the proof? statement 1. ng ≅ jg 2. ∠n and ∠j are right angles 3. ∠mgn ≅ ∠kgj 4. ∠n ≅ ∠j 5. △mng ≅ △kjg reason 1. given 2. given 3. vert ∠s are ≅ 4. rt. ∠s are ≅ 5. ? options: the reflexive property, asa, aas, the third angle theorem

Explanation:

• ASA (Angle - Side - Angle): This congruence criterion requires two angles and the included side of one triangle to be congruent to two angles and the included side of another triangle. But in our case, we have two angles (\(\angle MGN\cong\angle KGJ\), \(\angle N\cong\angle J\)) and a non - included side (\(\overline{NG}\cong\overline{JG}\)).
• AAS (Angle - Angle - Side): This congruence criterion 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 two triangles are congruent.
• We are given \(\overline{NG}\cong\overline{JG}\) (a side).
• \(\angle N\cong\angle J\) (an angle) and \(\angle MGN\cong\angle KGJ\) (another angle).
• Reflexive property: This property is about a figure being congruent to itself (\(a = a\)), which is not relevant here.
• Third - angle theorem: This theorem is about the third angle of two triangles being congruent if two pairs of angles are congruent, not a congruence criterion.

Answer:

AAS