To solve this problem, we need to find the measure of angle $\angle I$ in $\triangle HIJ$, where $HI \cong JH$ and $m \angle J = 34^\circ$.

Step-by-Step Solution

1. Identify Triangle Type: Since $HI \cong JH$, $\triangle HIJ$ is an isosceles triangle with $HI$ and $JH$ as the two equal sides.

2. Properties of Isosceles Triangles: In an isosceles triangle, the angles opposite the congruent sides are also congruent. So, $m \angle I = m \angle H$.

3. Sum of Angles in a Triangle: The sum of the angles in any triangle is $180^\circ$. Therefore, $m \angle H + m \angle J + m \angle I = 180^\circ.$

4. Substitute Known Values: We know $m \angle J = 34^\circ$, and $m \angle I = m \angle H$. Let $m \angle I = x$ and $m \angle H = x$. Thus, $x + 34^\circ + x = 180^\circ.$

5. Solve for $x$: $2x + 34^\circ = 180^\circ.$ $2x = 180^\circ - 34^\circ.$ $2x = 146^\circ.$ $x = 73^\circ.$

Therefore, $m \angle I = 73^\circ$.

Final Answer:

The measure of $\angle I$ is $73^\circ$.

Would you like further details or explanations on this solution?

Here are some related questions you may find helpful:

1. What are the properties of isosceles triangles?
2. How do you calculate the sum of angles in any triangle?
3. How would this problem change if the triangle was equilateral?
4. Can the same approach be used if $m \angle J$ was different?
5. How does this solution change if $\triangle HIJ$ was not isosceles?

Tip: Remember, in an isosceles triangle, the angles opposite the equal sides are always equal.