Let's solve the problems one by one.

Problem 10:

You are given:

• $m\angle 1 = x$
• $m\angle 2 = x - 6$

From the diagram, angles $\angle 1$ and $\angle 2$ are linear pair angles, meaning they are adjacent and their measures sum up to 180 degrees. The Linear Pair Theorem states that the sum of the measures of two angles that form a linear pair is 180 degrees.

So, we have the equation: $m\angle 1 + m\angle 2 = 180$ Substitute the given values: $x + (x - 6) = 180$ Simplify: $2x - 6 = 180$ Add 6 to both sides: $2x = 186$ Divide by 2: $x = 93$ Thus: $m\angle 1 = 93^\circ$ Substitute $x = 93$ into $m\angle 2 = x - 6$: $m\angle 2 = 93 - 6 = 87^\circ$

So, $m\angle 1 = 93^\circ$ and $m\angle 2 = 87^\circ$.

Justification: The Linear Pair Theorem.

Problem 11:

You are given:

• $m\angle 7 = 2x + 15$
• $m\angle 8 = 3x$

From the diagram, angles $\angle 7$ and $\angle 8$ are vertical angles, meaning they are congruent (have the same measure) because they are opposite each other when two lines intersect. The Vertical Angle Theorem states that vertical angles are congruent.

So, we have the equation: $m\angle 7 = m\angle 8$ Substitute the given values: $2x + 15 = 3x$ Subtract $2x$ from both sides: $15 = x$ Thus: $x = 15$ Now, substitute $x = 15$ into the expressions for $m\angle 7$ and $m\angle 8$: $m\angle 7 = 2(15) + 15 = 30 + 15 = 45^\circ$ $m\angle 8 = 3(15) = 45^\circ$

So, $m\angle 7 = 45^\circ$ and $m\angle 8 = 45^\circ$.

Justification: The Vertical Angle Theorem.

Would you like more details on any part of these solutions?

Here are 5 related questions to expand your understanding:

1. What is the definition of supplementary angles?
2. Can you identify other pairs of vertical angles in the diagram?
3. How would the solution change if $m\angle 1$ was given as $2x$ instead of $x$?
4. What is the significance of the Linear Pair Theorem in geometry?
5. How do you determine whether two angles are congruent?

Tip: Always look for relationships like vertical angles or linear pairs when dealing with intersecting lines—this can help you solve problems efficiently!