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### Problem 12:
- Given: \( m \angle LOJ = 8x \) and \( m \angle KOM = 2x + 30 \), and these angles form a linear pair (supplementary angles).
- Equation setup: 
  \[
  8x + (2x + 30) = 180^\circ
  \]
  Simplify:
  \[
  10x + 30 = 180
  \]
  \[
  10x = 150 \quad \Rightarrow \quad x = 15
  \]
  Now, find the individual angle measures:
  - \( m \angle KOM = 2x + 30 = 2(15) + 30 = 60^\circ \)
  - \( m \angle LOJ = 8x = 8(15) = 120^\circ \)
  - To verify: \( 120^\circ + 60^\circ = 180^\circ \), which is correct.

  **Answer:**
  - \( x = 15 \)
  - \( m \angle KOM = 60^\circ \)
  - \( m \angle LOJ = 120^\circ \)

### Problem 13:
- Given: \( \angle 1 \) and \( \angle 2 \) are complementary (add to 90°), and \( \angle 1 = x^2 - 5 \), \( \angle 2 = 4x \).
- The equation for complementarity is:
  \[
  (x^2 - 5) + 4x = 90
  \]
  Simplifying:
  \[
  x^2 + 4x - 95 = 0
  \]
  Solve using the quadratic formula: 
  \[
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-4 \pm \sqrt{16 + 380}}{2}
  \]
  \[
  x = \frac{-4 \pm \sqrt{396}}{2} = \frac{-4 \pm 19.9}{2}
  \]
  The solutions are \( x \approx 7.95 \) or \( x \approx -11.95 \). Since angles can't be negative, we choose \( x = 7.95 \).

  **Answer:** 
  - \( x \approx 7.95 \)
  - \( \angle 1 \approx (7.95^2 - 5) = 57.3^\circ \)
  - \( \angle 2 \approx 32.7^\circ \)
  
### Problem 14:
- Given a diagram with angles at a transversal intersection.
- \( m \angle 1 = 53^\circ \) (given directly).
- \( m \angle 2 = 37^\circ \) (vertical to angle marked 37°).
- \( m \angle 3 = 90^\circ \) (perpendicular intersection).
- \( m \angle 4 = 53^\circ \) (vertical to \( m \angle 1 \)).

  **Answer:** 
  - \( m \angle 1 = 53^\circ \)
  - \( m \angle 2 = 37^\circ \)
  - \( m \angle 3 = 90^\circ \)
  - \( m \angle 4 = 53^\circ \)

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Would you like to explore the details of these solutions further or go over specific steps? Here are five related questions to think about:

1. What is the general method to find the unknowns in a system of supplementary angles?
2. How do complementary angles work in geometric problems like problem 13?
3. What is the quadratic formula, and why is it used in problem 13?
4. Why are vertical angles always equal, as shown in problem 14?
5. How do we determine if an equation is solvable using the quadratic formula?

**Tip:** When solving geometry problems involving angles, always check if the angles form a linear pair or are complementary to ensure you're applying the correct angle relationships!

Find the measures of unknown angles using given relationships such as linear pairs, complementary angles, and vertical angles in the provided figures.

Learn how to solve angle problems involving linear pairs, complementary angles, and vertical angles. This guide also covers solving quadratic equations to find unknown angle measures. Suitable for students in Grades 8-10.

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