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In this diagram, we are given a kite \( GHIJ \), where \( \overline{GI} \) and \( \overline{HJ} \) are pairs of congruent adjacent sides, indicated by the tick marks. We're also provided with the following angle measures:

- \( \angle JGI = 121^\circ \)
- \( \angle GHI = 75^\circ \)

### Key properties of a kite:
1. **Adjacent sides are equal**: In a kite, two pairs of adjacent sides are congruent.
2. **Diagonals are perpendicular**: The diagonals of a kite intersect at right angles (90°).
3. **Two opposite angles are equal**: The angles between the non-congruent sides are congruent.

#### To find \( m\angle H \):

Since \( \angle GHI \) and \( \angle GJH \) form a pair of opposite angles in the kite, and the sum of all interior angles of a quadrilateral is \( 360^\circ \), we can use this fact to calculate \( \angle H \).

First, sum the given angles:
\[
\angle GJI = 121^\circ \quad \text{and} \quad \angle GHI = 75^\circ.
\]
Now, subtract the sum of these two angles from 360° to find the remaining sum for angles \( \angle H \) and \( \angle GJH \).

\[
360^\circ - (121^\circ + 75^\circ) = 360^\circ - 196^\circ = 164^\circ.
\]

Since \( \angle H = \angle GJH \) (as they are congruent opposite angles), divide the remaining sum by 2:
\[
\angle H = \frac{164^\circ}{2} = 82^\circ.
\]

Thus, \( m\angle H = 82^\circ \).

### Final Answer:
\[
m\angle H = 82^\circ
\]

Would you like further details or have any questions? Here are some related questions to explore:

1. How do we calculate the interior angles of any quadrilateral?
2. What makes the diagonals of a kite perpendicular?
3. How do opposite angles behave in a kite compared to other quadrilaterals?
4. How do you use congruency to simplify angle calculations?
5. What are other types of quadrilaterals with similar angle properties?

**Tip:** Always check if a quadrilateral has any specific properties like symmetry, congruent sides, or angle patterns that can simplify calculations.

Quadrilateral GHIJ is a kite. What is m∠H?

This problem involves finding the measure of angle H in the kite GHIJ, using the properties of a kite and the sum of interior angles in a quadrilateral. The problem is suitable for students in Grades 7-9 and covers key concepts such as congruent angles and kite geometry.

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