The problem states that BD bisects ∠ABC, which means that:

$\text{m∠ABD} = \text{m∠CBD}$

We are also given two expressions for the angles:

• $\text{m∠ABD} = 5x - 1$
• $\text{m∠CBD} = 2x + 5$

Since BD bisects ∠ABC, these two angles must be equal:

$5x - 1 = 2x + 5$

Step 1: Solve for $x$

Start by solving the equation $5x - 1 = 2x + 5$.

1. Subtract $2x$ from both sides: $5x - 2x - 1 = 5$ $3x - 1 = 5$

2. Add 1 to both sides: $3x = 6$

3. Divide by 3: $x = 2$

Step 2: Find the measure of ∠ABC

Since BD bisects ∠ABC, the measure of ∠ABC is the sum of the two equal angles ∠ABD and ∠CBD.

Now, substitute $x = 2$ into the expression for ∠ABD and ∠CBD:

• $\text{m∠ABD} = 5(2) - 1 = 10 - 1 = 9^\circ$
• $\text{m∠CBD} = 2(2) + 5 = 4 + 5 = 9^\circ$

Thus, $\text{m∠ABC} = \text{m∠ABD} + \text{m∠CBD} = 9^\circ + 9^\circ = 18^\circ$.

Final Answer:

• $x = 2$
• $\text{m∠ABC} = 18^\circ$

Would you like more details on the steps?

Here are 5 related questions:

1. How do you use angle bisectors to solve geometric problems?
2. What are some common properties of bisected angles in triangles?
3. Can this method be applied to non-bisected angles?
4. How would you solve a similar problem if it involved supplementary angles?
5. What happens if an exterior angle is bisected in a triangle?

Tip: Always check whether the sum of smaller angles equals the larger angle, especially when working with bisectors in geometry.