https://file.aiquickdraw.com/mathbot/file/172988647206567u0t09c.jpg

https://file.aiquickdraw.com/mathbot/file/1729886472066p6xz3ges.jpg

Let's solve the problems in the images.

### First Image

**Given:**
- \( m \angle DMC = 20^\circ \)
- \( m \angle FMD = 90^\circ \)
- \( m \angle EMC = 90^\circ \)

#### Find:
1. \( m \angle EMD \)

Since \( \angle FMD \) and \( \angle DMC \) form a right angle (90°) at \( M \), we can calculate \( m \angle EMD \) by adding the angles:

\[
m \angle EMD = 90^\circ - m \angle DMC = 90^\circ - 20^\circ = 70^\circ
\]

So, \( m \angle EMD = 70^\circ \).

2. \( m \angle DMB \)

Since \( m \angle EMC = 90^\circ \) and the angle at point \( M \) forms a straight line (180°), we can calculate \( m \angle DMB \):

\[
m \angle DMB = 180^\circ - m \angle EMC = 180^\circ - 90^\circ = 90^\circ
\]

So, \( m \angle DMB = 90^\circ \).

---

### Second Image

**Given:**
- \( m \angle EJI = 135^\circ \)

#### Find:
3. \( m \angle FJE \)

Since \( m \angle EJI = 135^\circ \) and \( \angle FJE \) forms a straight line with \( \angle EJI \), we can use the fact that straight angles sum to \( 180^\circ \):

\[
m \angle FJE = 180^\circ - m \angle EJI = 180^\circ - 135^\circ = 45^\circ
\]

So, \( m \angle FJE = 45^\circ \).

4. \( m \angle FJH \)

Here, \( m \angle FJH \) and \( m \angle FJE \) are vertically opposite angles, meaning they are equal:

\[
m \angle FJH = 45^\circ
\]

---

Let me know if you'd like more details or explanations. Here are some related questions to expand on this:

1. How do vertical angles help solve geometry problems?
2. What is the significance of supplementary angles in geometry?
3. Can all angles be calculated if one is given in a straight angle?
4. How do we apply the concept of perpendicular lines to solve angle problems?
5. What is the relationship between interior and exterior angles in polygons?

**Tip**: Always check for supplementary and vertical angles in geometry problems; they can simplify calculations greatly!

Solve for the missing angles in the given geometry diagrams.

Learn how to calculate angles in geometry problems using properties of straight angles and vertical angles. This solution shows how to apply the Angle Sum Property and Vertical Angle Theorem, suitable for Grades 7-9 students.

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