https://file.aiquickdraw.com/mathbot/file/17285685749011xz5ggz0.jpg

In the diagram, we are tasked with finding the measure of angle \( \angle HFJ \). Let's analyze the given information:

1. We have two triangles, \( \triangle GJF \) and \( \triangle HJF \), which appear to be congruent because they both have two equal sides marked as 5 units and a shared side \( JF \).
2. Angle \( \angle GJF \) is given as \( 38^\circ \), and both triangles have right angles at vertices \( G \) and \( H \) (since \( \angle GJH \) and \( \angle HJF \) are \( 90^\circ \)).

### Using the properties of triangles:

In \( \triangle GJF \), the sum of angles must equal \( 180^\circ \):
\[
\angle GJF + \angle GFJ + \angle JGF = 180^\circ
\]
Since \( \angle GJF = 38^\circ \) and \( \angle GJH = 90^\circ \), we can solve for \( \angle GFJ \):
\[
38^\circ + 90^\circ + \angle GFJ = 180^\circ
\]
\[
\angle GFJ = 180^\circ - 128^\circ = 52^\circ
\]

Because \( \triangle GJF \) and \( \triangle HJF \) are congruent, the angle \( \angle HFJ \) must be congruent to \( \angle GFJ \), which means \( \angle HFJ = 52^\circ \).

### Final answer:
\[
m\angle HFJ = 52^\circ
\]

Would you like further explanation on any part of this solution, or do you have any additional questions?

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Here are five related questions to explore further:
1. How do we know the triangles \( \triangle GJF \) and \( \triangle HJF \) are congruent?
2. What are the properties of congruent triangles, and how are they useful here?
3. How would the solution change if the angle \( \angle GJF \) were different?
4. What are the implications of the right angles at \( G \) and \( H \) for the shape of the figure?
5. How can we verify the lengths of the sides in both triangles?

**Tip:** Always double-check the angle sum in a triangle, which must always add up to \( 180^\circ \).

Find the measure of angle HFJ in the given diagram with two right triangles, both having two sides of length 5 and one angle marked as 38°.

Learn how to find the measure of angle HFJ in a geometry problem involving congruent triangles and the angle sum property. The problem provides two right triangles with equal side lengths and an angle of 38°, suitable for students in Grades 8-10.

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