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This problem involves finding the measures of various angles in a geometric figure, using the given values of certain angles and the relationships between them. Here's a step-by-step breakdown:

### Given:
- \( m\angle 2 = 41^\circ \)
- \( m\angle 5 = 94^\circ \)
- \( m\angle 10 = 109^\circ \)

We are asked to find the following angle measures:
- \( m\angle 1 \)
- \( m\angle 3 \)
- \( m\angle 4 \)
- \( m\angle 6 \)
- \( m\angle 7 \)
- \( m\angle 8 \)
- \( m\angle 9 \)

### Approach:

#### a. \( m\angle 1 \):
Angle \( \angle 1 \) and \( \angle 2 \) are adjacent to form a straight line, making them supplementary. Thus:
\[
m\angle 1 + m\angle 2 = 180^\circ
\]
\[
m\angle 1 = 180^\circ - 41^\circ = 139^\circ
\]

#### b. \( m\angle 3 \):
Angle \( \angle 3 \) is vertically opposite to \( \angle 2 \), so they are equal:
\[
m\angle 3 = m\angle 2 = 41^\circ
\]

#### c. \( m\angle 4 \):
Angles \( \angle 3 \), \( \angle 4 \), and \( \angle 5 \) form a triangle. The sum of the angles in a triangle is \( 180^\circ \):
\[
m\angle 3 + m\angle 4 + m\angle 5 = 180^\circ
\]
Substitute the known values:
\[
41^\circ + m\angle 4 + 94^\circ = 180^\circ
\]
\[
m\angle 4 = 180^\circ - 135^\circ = 45^\circ
\]

#### d. \( m\angle 6 \):
Angle \( \angle 6 \) and \( \angle 5 \) form a linear pair, meaning they are supplementary:
\[
m\angle 6 + m\angle 5 = 180^\circ
\]
\[
m\angle 6 = 180^\circ - 94^\circ = 86^\circ
\]

#### e. \( m\angle 7 \):
Angle \( \angle 7 \) is vertically opposite to \( \angle 6 \), so:
\[
m\angle 7 = m\angle 6 = 86^\circ
\]

#### f. \( m\angle 8 \):
Angles \( \angle 8 \), \( \angle 9 \), and \( \angle 10 \) form a triangle, so their sum must be \( 180^\circ \):
\[
m\angle 8 + m\angle 9 + m\angle 10 = 180^\circ
\]
But we don't know \( m\angle 9 \) yet, so we'll need to solve for it later.

#### g. \( m\angle 9 \):
Angle \( \angle 9 \) and \( \angle 8 \) form a linear pair, meaning they are supplementary:
\[
m\angle 9 + m\angle 8 = 180^\circ
\]
At this stage, we will need more information about either \( \angle 8 \) or \( \angle 9 \) to calculate.

Let me know if you would like further assistance or have more specific questions on this. Here are some related problems:

1. How do you find the third angle in a triangle when two angles are known?
2. What is the relationship between vertically opposite angles?
3. How do supplementary angles relate to each other?
4. How can you use triangle angle sums to solve for unknown angles?
5. How do you find angle measures for a quadrilateral?

**Tip:** In geometry problems involving lines and angles, always look for linear pairs and vertical angles to simplify your calculations.

If m∠2 = 41°, m∠5 = 94°, and m∠10 = 109°, find each measure.

Discover how to solve for angles in a complex geometric figure using given angle measures and relationships. This guide covers the calculations for supplementary, vertical, and triangle angle sums, suitable for students in Grades 6-8.

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