Let's go through this problem step by step. The problem states:

1. Given:

   • $m\angle 7 = 53^\circ$
   • $m\angle 6 = (10x + 7)^\circ$

2. Goal:

   • Determine the measures of the other seven angles in the figure.

Step 1: Solve for $x$

Since $m$ and $n$ are parallel lines, angles 6 and 7 are alternate interior angles. Therefore, they are congruent, meaning: $m\angle 6 = m\angle 7$ Thus, $10x + 7 = 53$

Subtract 7 from both sides: $10x = 46$

Now, divide by 10: $x = 4.6$

Step 2: Substitute $x$ to Find $m\angle 6$

Now that we have $x = 4.6$, substitute it back into the expression for $m\angle 6$: $m\angle 6 = 10(4.6) + 7 = 46 + 7 = 53^\circ$

So, $m\angle 6 = 53^\circ$, which matches $m\angle 7$ as expected.

Step 3: Determine the Other Angles Using Angle Relationships

Since angles $m\angle 6$ and $m\angle 7$ are both $53^\circ$, we can use the properties of parallel lines to find the measures of all other angles:

• Angles 6, 7, 2, and 3 are all corresponding or alternate interior angles, so they are each $53^\circ$.
• Angles 1, 4, 5, and 8 are the supplementary angles to these (since each pair adds up to $180^\circ$).

Thus: $m\angle 1 = m\angle 4 = m\angle 5 = m\angle 8 = 180^\circ - 53^\circ = 127^\circ$

Summary of Angle Measures

• $m\angle 1 = 127^\circ$
• $m\angle 2 = 53^\circ$
• $m\angle 3 = 53^\circ$
• $m\angle 4 = 127^\circ$
• $m\angle 5 = 127^\circ$
• $m\angle 6 = 53^\circ$
• $m\angle 7 = 53^\circ$
• $m\angle 8 = 127^\circ$

Let me know if you'd like further details or have any questions.

Questions to Expand Understanding

1. How do alternate interior angles help in solving problems with parallel lines?
2. Why are corresponding angles equal in parallel line configurations?
3. How do supplementary angles assist in finding unknown angle measures?
4. What other methods can be used to find missing angles when dealing with parallel lines?
5. Can we apply these principles to non-parallel line problems?

Tip

Remember, whenever you see parallel lines cut by a transversal, think of corresponding, alternate interior, and supplementary angles to determine unknown angles.