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Let's break down the problem from the image you've uploaded.

We are given:
- \( \angle DEF = 47^\circ \)
- \( \angle HFG = 88^\circ \)

We are asked to find:
a. \( \angle DGH \)
b. \( \angle FHG \)
c. \( \angle EDG \)

### Steps to solve:

#### a. Find \( \angle DGH \):
Notice that the quadrilateral \( DEFG \) is inscribed in the circle. Therefore, opposite angles of an inscribed quadrilateral sum up to \( 180^\circ \). Since \( \angle DEF = 47^\circ \), the opposite angle, \( \angle DGH \), is:
\[
\angle DGH = 180^\circ - 47^\circ = 133^\circ
\]

#### b. Find \( \angle FHG \):
We are given \( \angle HFG = 88^\circ \), and the two angles \( \angle HFG \) and \( \angle FHG \) form a linear pair (they are on a straight line), so they sum to \( 180^\circ \). Hence:
\[
\angle FHG = 180^\circ - 88^\circ = 92^\circ
\]

#### c. Find \( \angle EDG \):
Since \( \angle DEF = 47^\circ \) and both \( D \), \( E \), and \( G \) lie on the circle, we can use the property of inscribed angles. \( \angle EDG \) is subtended by the same arc as \( \angle DEF \). Therefore:
\[
\angle EDG = \angle DEF = 47^\circ
\]

### Summary of answers:
a. \( \angle DGH = 133^\circ \)  
b. \( \angle FHG = 92^\circ \)  
c. \( \angle EDG = 47^\circ \)

Would you like more details on any part of this? Here are five related questions to deepen your understanding:

1. How do we determine the sum of opposite angles in an inscribed quadrilateral?
2. What is the property of linear pairs of angles?
3. How do inscribed angles relate to the arcs they subtend?
4. Why do opposite angles in a cyclic quadrilateral sum to \( 180^\circ \)?
5. Can this method be applied to any quadrilateral inscribed in a circle?

**Tip**: Always remember that the sum of angles around a point on a straight line is \( 180^\circ \), and the sum of opposite angles in a cyclic quadrilateral is also \( 180^\circ \).

Given a circle with points D, E, F, G, and H such that ∠DEF = 47° and ∠HFG = 88°, determine the measures of angles ∠DGH, ∠FHG, and ∠EDG.

Explore how to find angles ∠DGH, ∠FHG, and ∠EDG in a cyclic quadrilateral inscribed in a circle with given angles ∠DEF and ∠HFG. This problem leverages key theorems such as the Inscribed Quadrilateral Theorem and the Linear Pair Postulate, suitable for high school geometry students.

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