In triangle, I, J, K, comma△IJK, start overline, J, K, end overline, \cong, start overline, I, J, end overline 
JK
 ≅ 
IJ
  and m, angle, I, equals, 23, degrees, .m∠I=23 
∘
 . Find m, angle, J, .m∠J.

In the triangle \( \triangle IJK \), we are given that:

- \( \overline{JK} \cong \overline{IJ} \), meaning the sides \( JK \) and \( IJ \) are congruent.
- \( m\angle I = 23^\circ \).

Since \( \overline{JK} \cong \overline{IJ} \), the triangle \( \triangle IJK \) is **isosceles**, meaning the base angles are equal. The base angles in this case are \( \angle J \) and \( \angle K \).

The sum of the interior angles in any triangle is \( 180^\circ \). So, we can write the following equation for the sum of the angles in the triangle:

\[
m\angle I + m\angle J + m\angle K = 180^\circ
\]

Since \( \triangle IJK \) is isosceles, we know that \( m\angle J = m\angle K \). Let \( x \) represent the measure of both \( \angle J \) and \( \angle K \). Therefore, the equation becomes:

\[
23^\circ + x + x = 180^\circ
\]

Simplifying:

\[
23^\circ + 2x = 180^\circ
\]

Subtract \( 23^\circ \) from both sides:

\[
2x = 157^\circ
\]

Now, divide by 2:

\[
x = 78.5^\circ
\]

Thus, \( m\angle J = 78.5^\circ \).

### Final answer:
\[
m\angle J = 78.5^\circ
\]

Would you like further clarification or more details on any step?

Here are 5 related questions to expand your understanding:
1. How would the solution change if \( \triangle IJK \) were not isosceles?
2. What are the properties of isosceles triangles that were used in solving this problem?
3. Can you generalize the method for finding the angles in any isosceles triangle?
4. What is the relationship between the base angles in an isosceles triangle?
5. How does the sum of interior angles apply to other types of polygons?

**Tip:** In any triangle, the sum of the interior angles is always \( 180^\circ \), regardless of its type!

In triangle △IJK, JK ≅ IJ and m∠I = 23°. Find m∠J.

Learn how to find m∠J in isosceles triangle △IJK, where sides JK ≅ IJ and m∠I = 23°. This geometry problem applies the angle sum property and isosceles triangle theorem to solve for m∠J. Suitable for students in Grades 6-8.

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