https://file.aiquickdraw.com/mathbot/file/1726650077689f66ublw2.jpg

Let's go through the questions in the diagram one by one. I will explain the reasoning and solutions.

### Question 3.1:
**Lines BG and CF never cross or intersect. What is the equation for line CF?**

- From the diagram, we can observe that line \( CF \) is parallel to line \( BG \).
- The equation for line \( BG \) is already given as \( y = 2x + 4 \). Since \( CF \) is parallel to \( BG \), the slope of \( CF \) must also be the same as \( BG \), i.e., slope \( m = 2 \).
- To find the equation of line \( CF \), we need to find the y-intercept, denoted as \( c \).
- Point \( F \) lies on line \( CF \) and has coordinates \( F(x, y) = (6, 7) \) based on the grid.
  
Using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = 2 \), \( x_1 = 6 \), and \( y_1 = 7 \):
\[
y - 7 = 2(x - 6)
\]
\[
y - 7 = 2x - 12
\]
\[
y = 2x - 12 + 7
\]
\[
y = 2x - 5
\]
Thus, the equation of line \( CF \) is \( \boxed{y = 2x - 5} \).

---

### Question 3.2:
**What is the size of angle HIG?**

- From the diagram, we can observe that \( HIG \) is a straight angle (a line through points \( H, I, G \)), and a straight line forms an angle of \( 180^\circ \).
Thus, the size of angle \( HIG \) is \( \boxed{180^\circ} \).

---

### Question 3.3:
**What is the size of angle DLE?**

- From the diagram, angle \( DLE \) and angle \( MLF \) form a linear pair (supplementary angles), and the sum of supplementary angles is \( 180^\circ \).
- We are given that \( \angle MLF = 137^\circ \).
Thus:
\[
\angle DLE = 180^\circ - 137^\circ = 43^\circ
\]
So, the size of angle \( DLE \) is \( \boxed{43^\circ} \).

---

### Question 3.4:
**How does the size of angle IJK relate to the size of angle MKL?**

- From the diagram, it appears that lines \( IJ \) and \( KL \) are intersected by line \( JK \), forming a pair of alternate interior angles.
- Since alternate interior angles are congruent when lines are parallel, the size of angle \( IJK \) is equal to the size of angle \( MKL \).
Thus:
\[
\angle IJK = \boxed{\angle MKL}
\]

---

### Question 3.5:
**If MK = 3 meters and KL = 4 meters, then how long is LM?**

- From the diagram, we can see that triangle \( MKL \) forms a right triangle.
- By applying the Pythagorean Theorem:
\[
LM^2 = MK^2 + KL^2
\]
\[
LM^2 = 3^2 + 4^2 = 9 + 16 = 25
\]
\[
LM = \sqrt{25} = 5 \, \text{meters}
\]
Thus, the length of \( LM \) is \( \boxed{5} \, \text{meters} \).

---

### Question 3.6:
**What is the sum of angle MIJ plus angle IJK plus angle KLM?**

- \( \angle MIJ \), \( \angle IJK \), and \( \angle KLM \) are the angles around point \( J \) and along a straight line, so the sum of these angles should equal \( 180^\circ \).
Thus:
\[
\boxed{180^\circ}
\]

---

Would you like more details on any of the steps, or do you have any further questions?

### 5 Related Questions:
1. How do we calculate the equation of a line if it passes through two given points?
2. What is the general form of the equation of a straight line?
3. Can you explain how to determine congruent angles in different types of geometrical figures?
4. How does the Pythagorean Theorem help in solving distances in geometry?
5. What are supplementary angles, and how are they related to linear pairs?

### Tip:
When working with parallel lines and transversals, always look for corresponding and alternate angles, as they are congruent, making many problems easier to solve.

Lines BG and CF never cross or intersect. What is the equation for line CF? Show your work or explain your reasoning. (5)

In this problem, you will learn how to find the equation of line CF, parallel to line BG, using the slope and point-slope form. You'll also solve for various angles using the Pythagorean theorem and angle properties. Suitable for students in Grades 9-11.

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