The problem states:

• $m\angle GEF$ is thirteen less than five times $m\angle DEG$.
• $m\angle DEF = 149^\circ$.

We are tasked with finding $m\angle GEF$.

Let's solve this step by step:

Step 1: Relationship Between Angles

The angle $\angle DEF$ is given as $149^\circ$. Notice that the figure suggests a straight line at point $E$ along the line segment $EF$. This means the sum of the angles $\angle GEF + \angle DEG + \angle DEF = 180^\circ$ (as the sum of angles on a straight line is $180^\circ$).

Thus, we have: $m\angle GEF + m\angle DEG + 149^\circ = 180^\circ$ $m\angle GEF + m\angle DEG = 31^\circ \quad \text{(subtracting 149 from both sides)}$

Step 2: Express $m\angle GEF$ in Terms of $m\angle DEG$

According to the problem, $m\angle GEF = 5 \cdot m\angle DEG - 13$.

Step 3: Substitute the Relationship

We now substitute $m\angle GEF = 5 \cdot m\angle DEG - 13$ into the equation from Step 1: $(5 \cdot m\angle DEG - 13) + m\angle DEG = 31^\circ$ Simplifying: $5 \cdot m\angle DEG + m\angle DEG - 13 = 31^\circ$ $6 \cdot m\angle DEG - 13 = 31^\circ$ Add 13 to both sides: $6 \cdot m\angle DEG = 44^\circ$ Divide by 6: $m\angle DEG = \frac{44^\circ}{6} = 7.33^\circ$

Step 4: Find $m\angle GEF$

Now, substitute $m\angle DEG = 7.33^\circ$ into the expression for $m\angle GEF$: $m\angle GEF = 5 \cdot 7.33^\circ - 13 = 36.65^\circ - 13 = 23.65^\circ$

Final Answer:

$m\angle GEF \approx 23.65^\circ$.

Let me know if you'd like more details or have any questions! Here are some follow-up questions to deepen your understanding:

1. How do you identify angle relationships in geometric figures like this one?
2. Why is the sum of angles on a straight line always $180^\circ$?
3. What is the impact of solving for one angle using relationships with others?
4. How do systems of equations help in finding unknown angles?
5. Can you explain how the information given in word problems translates to equations?

Tip: Always start by writing down what is given and any geometric relationships (like angles on a straight line) to systematically approach the problem.