https://file.aiquickdraw.com/mathbot/file/1726650391345fkp9931a.jpg

Let's tackle these questions one by one based on the provided diagram.

### 3.1 Lines BG and CF never cross or intersect. What is the equation for line CF?
Given that line BG is \( y = 2x + 4 \), and BG and CF are parallel, CF will have the same slope as BG. The slope is 2. To find the y-intercept for CF, we observe that it passes through point D, located at (4, 0). We use the point-slope form of the line equation:

\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is (4, 0) and \( m = 2 \):

\[ y - 0 = 2(x - 4) \]
\[ y = 2x - 8 \]

Therefore, the equation for line CF is \( y = 2x - 8 \).

### 3.2 What is the size of angle HIG?
Since line FG is parallel to the x-axis, \( \angle HIG \) would be the same as the angle of inclination of line HI to the x-axis, which is parallel to BG and CF. The angle of inclination is derived from the slope, which is 2 (as \( \tan(\theta) = 2 \)). We can calculate \( \theta \) as:
\[ \theta = \tan^{-1}(2) \]
Let's calculate this angle:

### Calculating angle HIG
Using the arctangent function, we'll find \( \theta \) from the slope of 2. Let's calculate this.The size of angle \( HIG \) is approximately \( 63.43^\circ \).

Do you need any further explanation on these or shall we proceed to the next questions? Here are some follow-up topics you might consider:

1. How to find the slope of a line from its equation.
2. The concept of parallel lines and their properties.
3. How to use point-slope form to find the equation of a line.
4. Understanding the tangent function and its role in angle calculations.
5. Exploring different ways to measure angles in geometry.

Tip: When checking if lines are parallel, compare their slopes. If they are the same, the lines do not intersect and are parallel.

Based on the provided diagram, solve the following:
3.1 Lines BG and CF never cross or intersect. What is the equation for line CF?
3.2 What is the size of angle HIG?
3.3 What is the size of angle DLE?
3.4 How does the size of angle IJK relate to the size of angle MKL?
3.5 If MK = 3 meters and KL = 4 meters, then how long is LM?
3.6 What is the sum of angle MIJ plus angle IJK plus angle KLM?

Solve a geometry problem involving parallel lines BG and CF, using slope-intercept form and trigonometric functions to calculate angles HIG and DLE. Understand the relationship between angles IJK and MKL. This problem is suitable for high school students (Grades 10-12).

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