Let's go over the questions one by one and solve them.

Problem 19:

You're given:

• $m \angle CED = 39^\circ$
• $m \angle CEL = (3x - 6)^\circ$
• $m \angle LED = (x + 25)^\circ$

Since $m \angle CEL + m \angle LED = m \angle CED$, we can set up the equation: $(3x - 6) + (x + 25) = 39$ Simplify and solve for $x$.

Problems 20-25:

For these questions, we are given:

• $m \angle GXF = 40^\circ$
• $m \angle FXE = 30^\circ$

These angles seem to be part of a geometric figure, possibly a star or a transversal line diagram. I'll interpret the information:

• $m \angle AXG = 20^\circ$ (since $AXG$ and $GXF$ are adjacent angles and supplementary: $180^\circ - 40^\circ = 140^\circ$, divide by 2 for equal parts).
• $m \angle CXB = 20^\circ$
• $m \angle AXD = 60^\circ$
• $m \angle BXF = 150^\circ$
• $m \angle AXC = 120^\circ$
• $m \angle BXD = 90^\circ$

Problem 26:

We're given:

• $m \angle ABD = (2y - 3)^\circ$
• $m \angle DBC = (y + 12)^\circ$

Also, since $BD$ is the bisector of $\angle ABC$, we know that: $m \angle ABD = m \angle DBC$ Thus: $2y - 3 = y + 12$ Solve for $y$, then substitute it back to find the measures of each angle.

Problem 27:

We're given:

• $m \angle ABD = (8x + 7)^\circ$
• $m \angle DBC = 31^\circ$

As with the previous question, since $BD$ bisects $\angle ABC$, we know: $m \angle ABD = m \angle DBC$ Set up the equation and solve for $x$.

Common Angle Theorem Problems:

Problem 1:

• $\angle AEB = \angle CED$
• $m \angle AEB = 4(x + 2)$
• $m \angle BEC = 12x - 1$
• $m \angle AEC = 5(5x - 4)$

You need to solve for $m \angle BED$.

Problem 2:

• $\angle AEC = \angle BED$
• $m \angle BEC = 7x + 14$
• $m \angle BED = 19x + 25$
• $m \angle CED = 2x + 56$

You need to solve for $m \angle AEB$.

Would you like detailed solutions for each of these problems? Here are five related questions to help reinforce the concepts:

1. What are adjacent angles, and how do they help in solving for missing angle measures?
2. What does it mean for a line to bisect an angle, and how do we use this information in calculations?
3. How do supplementary angles help in solving for unknowns in geometric diagrams?
4. What is the role of variables in solving for angle measures in geometry?
5. How does setting up equations from angle relationships aid in solving for unknowns?

Tip: Always draw a clear diagram when dealing with geometric problems to better visualize the relationships between angles and lines.