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Let's break down the solution step by step:

The kite has four vertices: \( D \), \( E \), \( F \), and \( G \). The given angles are:
- \( \angle GFD = 84^\circ \)
- \( \angle DFE = 138^\circ \)

We need to find \( \angle EFG \).

### Key properties of a kite:
1. **Symmetry**: The diagonals of a kite intersect at right angles.
2. **Angle Sum**: The sum of the interior angles of a quadrilateral is always \(360^\circ\).

So for the quadrilateral \( DEFG \), the sum of the angles is:
\[
\angle GFD + \angle DFE + \angle FEG + \angle EFG = 360^\circ
\]

### Step-by-step solution:
- We know \( \angle GFD = 84^\circ \) and \( \angle DFE = 138^\circ \).
- Let’s call \( \angle EFG = x \).
- The other angle, \( \angle FEG \), will be the same as \( \angle GFD \) because the kite is symmetrical, so \( \angle FEG = 84^\circ \).

Now we can use the angle sum property:
\[
84^\circ + 138^\circ + 84^\circ + x = 360^\circ
\]
Simplify:
\[
306^\circ + x = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 306^\circ = 54^\circ
\]

Therefore, \( \angle EFG = 54^\circ \).

---

Would you like more details on this solution or have any questions? Here are 5 related questions to consider:

1. What are the properties of a kite in geometry?
2. How do you calculate the sum of interior angles in a quadrilateral?
3. Can a kite have more than one pair of congruent angles?
4. How are the diagonals of a kite related to its angles?
5. How does symmetry help in solving kite-related problems?

**Tip**: Always remember that the sum of interior angles of any quadrilateral is \(360^\circ\).

Look at the kite below. Calculate the size of angle EFG. Give your answer in degrees (°).

Learn how to calculate the size of angle EFG in a kite, using key geometry concepts such as the sum of the interior angles of a quadrilateral and the symmetry of a kite. This step-by-step solution is suitable for students in Grades 6-8.

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