https://file.aiquickdraw.com/mathbot/file/17269331230223c49vtai.jpg

Let's break down the problem from the image:

The given figure has the following information:
- \( \triangle ABC \) is an equilateral triangle.
- Lines \( DIP \) and \( IGE \) are straight.
- \( IHF \parallel BPC \).
- \( \angle AIG = \frac{\angle EFT}{2} \), and we know \( x - y = 80^\circ \).

### Goal:
We need to find the measures of \( \angle DAI \) and \( \angle GIH \).

### Step-by-Step Breakdown:

#### 1. Analyze the Equilateral Triangle
Since \( \triangle ABC \) is equilateral, all its internal angles are \( 60^\circ \). Hence:
- \( \angle BAC = 60^\circ \)
  
#### 2. Use the Relation \( x - y = 80^\circ \)
Given \( x - y = 80^\circ \), we can infer that \( x \) and \( y \) are angles related to the lines \( DIP \) and \( IGE \). We'll need more geometric relations to determine \( x \) and \( y \).

#### 3. Parallel Lines
The fact that \( IHF \parallel BPC \) suggests that we can use properties of alternate interior angles and corresponding angles in these two parallel lines to solve for angles around point \( A \), especially with respect to \( \angle AIG \).

#### 4. Use \( \angle AIG = \frac{\angle EFT}{2} \)
We are also given a relationship between \( \angle AIG \) and \( \angle EFT \). This will help in calculating \( \angle AIG \), but we need more details about \( \angle EFT \), likely related to \( \angle GIH \) or the configuration around point \( F \).

To fully solve the problem and get exact values for \( \angle DAI \) and \( \angle GIH \), we'll need to make some calculations based on the given angle relationships and potentially use the properties of parallel lines and equilateral triangles. 

I'll continue working on this and get back with the exact calculations.

In the given figure, ABC is an equilateral triangle, DIP and IGE are straight lines. If IHF || BPC, ∠AIG = ∠EFT / 2 and x - y = 80°, then find the measures of ∠DAI and ∠GIH.

This problem involves solving for angles in a figure where ABC is an equilateral triangle and parallel lines are involved. We use properties of triangles, parallel lines, and the relationships between angles to find the measures of ∠DAI and ∠GIH.

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