To solve for the measure of angle 7 based on the given diagram, here's the approach:

Given Information:

• Angle 1: $3x + 10^\circ$
• Angle 4: $4x - 16^\circ$

The goal is to find the measure of angle 7.

Analysis:

This is a diagram involving two parallel lines ($a$ and $b$) intersected by two transversal lines ($c$ and a vertical transversal). Angles 1 and 4 are related by corresponding angle relationships and other geometric properties.

Key Angle Relationships:

• Corresponding angles are equal: $\angle 1 = \angle 5$, $\angle 4 = \angle 8$, etc.
• Alternate interior angles are equal: $\angle 1 = \angle 4$, $\angle 2 = \angle 3$, etc.
• Vertical angles are equal: $\angle 1 = \angle 3$, $\angle 4 = \angle 2$, etc.

Since angles 1 and 4 are alternate interior angles, they are equal: $\text{Angle } 1 = \text{Angle } 4$

Thus, we can set up the equation: $3x + 10^\circ = 4x - 16^\circ$

Solve for $x$:

Let's solve for $x$ first.

$3x + 10 = 4x - 16$ Subtract $3x$ from both sides: $10 = x - 16$ Add 16 to both sides: $x = 26$

Substitute $x = 26$ into the angle expressions:

1. Angle 1: $3x + 10 = 3(26) + 10 = 78 + 10 = 88^\circ$
2. Angle 4: $4x - 16 = 4(26) - 16 = 104 - 16 = 88^\circ$

Both angles 1 and 4 are $88^\circ$.

Finding Angle 7:

Since angles 4 and 7 are corresponding angles, they are equal. Therefore, the measure of angle 7 is also $88^\circ$.

Conclusion:

The measure of angle 7 is $88^\circ$.

Would you like further details or have any additional questions?

Here are some related questions to consider:

1. What is the sum of angles around a point formed by two intersecting lines?
2. How do corresponding angles work in parallel lines and transversals?
3. What is the relationship between alternate interior angles in parallel lines?
4. Can we calculate angle 2 using the information given?
5. How would we verify that the lines are indeed parallel?

Tip: When working with parallel lines and transversals, corresponding angles are always congruent. This can simplify solving for unknown angles.