We are given that $\angle A$ is a right angle, $DC = GC$, and $TO = TG$. We need to find $m \angle DGO$.

Step-by-Step Solution:

1. Recognize the Symmetry:

   • Since $DC = GC$ and $TO = TG$, we know that both triangle $\triangle DCG$ and triangle $\triangle TOG$ are isosceles triangles.

2. Angles in Triangle:

   • Since $\triangle ADC$ is a right triangle with $\angle A = 90^\circ$, we can conclude that $\triangle ADC$ is a right-angled triangle.
   • Similarly, since $\triangle ATG$ also shares a common right-angle at $\angle A$, $\triangle ATG$ is also a right-angled triangle.

3. Triangle Relations:

   • In the isosceles triangle $\triangle DCG$, we know that the two angles at $D$ and $G$ must be equal because $DC = GC$.
   • Similarly, in the isosceles triangle $\triangle TOG$, the angles at $O$ and $G$ must also be equal because $TO = TG$.

4. Finding $\angle DGO$:

   • $\triangle DGO$ is a triangle formed by the points $D$, $G$, and $O$.
   • Using the symmetry and angle relationships in the isosceles triangles mentioned, and considering that the sum of the interior angles of any triangle is $180^\circ$, we conclude that $\angle DGO$ must be 90 degrees.

Answer:

$m \angle DGO = 90^\circ$.

Would you like further details on this solution, or have any questions?

Here are 5 related questions to consider:

1. What properties define an isosceles triangle?
2. How do you calculate the sum of angles in any triangle?
3. What is the significance of symmetry in geometry?
4. Can two right triangles be congruent even if they are not similar?
5. How can angle bisectors help in finding unknown angles?

Tip: In any geometric problem involving isosceles triangles, always check if the two equal sides lead to equal angles. This symmetry can simplify finding unknown angles.