https://file.aiquickdraw.com/mathbot/file/1726079350752alqge6hr.jpg

This diagram shows a triangle \( LMN \) with two other angles labeled as expressions involving \( k \). Specifically, the angles are given as:

- \( \angle ZLM = (4k + 5)^\circ \)
- \( \angle LNM = (6k + 10)^\circ \)
- \( \angle MYN = 115^\circ \)

The sum of the angles in any triangle must equal \( 180^\circ \). We can use this fact to set up an equation to solve for \( k \).

### Step-by-step solution:

The three angles of triangle \( LMN \) are:

\[
(4k + 5)^\circ + (6k + 10)^\circ + 115^\circ = 180^\circ
\]

Simplifying the equation:

\[
(4k + 6k + 5 + 10 + 115) = 180
\]
\[
10k + 130 = 180
\]

Subtracting 130 from both sides:

\[
10k = 50
\]

Dividing both sides by 10:

\[
k = 5
\]

Now, we can substitute \( k = 5 \) back into the expressions for the angles:

- \( \angle ZLM = 4k + 5 = 4(5) + 5 = 20 + 5 = 25^\circ \)
- \( \angle LNM = 6k + 10 = 6(5) + 10 = 30 + 10 = 40^\circ \)

So, the angles of the triangle are:

- \( \angle ZLM = 25^\circ \)
- \( \angle LNM = 40^\circ \)
- \( \angle MYN = 115^\circ \)

The solution satisfies the condition that the sum of the angles equals 180°.

Would you like more details or further clarification?

---

Here are five related questions you might consider:

1. How do we know that the sum of the angles in a triangle always equals 180°?
2. Can this method be used for non-Euclidean triangles?
3. What is the importance of the external angle \( \angle MYN \)?
4. What would happen if one of the angles were given in radians instead of degrees?
5. Could this problem be solved graphically or by using a geometric tool?

**Tip:** Always check the sum of the angles in any triangle when solving for variables to ensure the solution is consistent.

Find the value of k based on the angles of triangle LMN where the angles are given as (4k + 5)°, (6k + 10)°, and 115°.

Learn how to solve for the variable k in a triangle where the angles are given as (4k + 5)°, (6k + 10)°, and 115°. This problem involves algebra and the triangle angle sum theorem, suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation