Question

The proof that TriangleMNG ≅ TriangleKJG is shown.

Given: AngleN and AngleJ are right angles; NG ≅ JG
Prove: TriangleMNG ≅ TriangleKJG

Triangles M N G and J K G share point G. Angles M N G and G J K are right angles. Sides N G and G J are congruent.
What is the missing reason in the proof?

A 2-column table has 5 rows. Column 1 is labeled Statement with entries line segment N G is-congruent-to line segment J G, angle N and angle J are right angles, angle M G N is-congruent-to angle K G J, angle N is-congruent-to angle J, triangle M N G is-congruent-to triangle K J G. Column 2 is labeled reason with entries given, given, vertical angles are congruent, right angles are congruent, and question mark.
the reflexive property
ASA
AAS
the third angle theorem

Answer 1

To prove that Triangle MNG is congruent to Triangle KJG, we can use the following information given in the proof:

1. Both triangles have a right angle (Angle N and Angle J).
2. The segments NG and JG are congruent (NG ≅ JG).
3. We have established that angle MGN is congruent to angle KGJ (vertical angles) and that angle N is congruent to angle J.

Since both triangles share point G, we can conclude that angle MGN and angle KGJ are also congruent because they are vertical angles. This gives us two angles and the side between them (NG ≅ JG):

- We have angle MNG (the right angle), angle MGN (congruent to angle KGJ), and side NG (congruent to JG) forming the two shapes.

Thus, the missing reason in the last row of the proof is **ASA (Angle-Side-Angle)**, as we have two angles and the included side that are congruent.

So the answer is:
**ASA**.