Math Problem Statement

O is on the interior of ∠LMN. If m∠LMO = 2x + 5 and m∠NMO = 57°, m∠LMN = 13x - 10, which equation could be used to find the value of x?

Solution

The problem involves angles within a triangle. Specifically, we are given the measures of three angles and need to set up an equation to solve for xx.

Given Information:

  • OO is a point on the interior of LMN\triangle LMN.
  • mLMO=2x+5m \angle LMO = 2x + 5
  • mNMO=57m \angle NMO = 57^\circ
  • mLMN=13x10m \angle LMN = 13x - 10

Key Concept:

In any triangle, the sum of the angles is always 180180^\circ. Therefore, we can set up an equation based on this fact: mLMO+mNMO+mLMN=180m \angle LMO + m \angle NMO + m \angle LMN = 180^\circ Substitute the given angle expressions: (2x+5)+57+(13x10)=180(2x + 5) + 57 + (13x - 10) = 180 Simplify the equation: 2x+5+57+13x10=1802x + 5 + 57 + 13x - 10 = 180 15x+52=18015x + 52 = 180 Now, solve for xx: 15x=1805215x = 180 - 52 15x=12815x = 128 x=12815x = \frac{128}{15} x8.53x \approx 8.53

Thus, the correct equation to solve for xx is: 15x+52=18015x + 52 = 180 This corresponds to the third option in the list:

  • 15x5=5715x - 5 = 57.

Would you like further details on the solution or clarification? Here are some related questions:

  1. How do we verify if the solution for xx satisfies all the given angles?
  2. What is the sum of the angles in a polygon, and how does it generalize for more sides?
  3. Can you explain why the sum of angles in a triangle is always 180180^\circ?
  4. How can we check if a given triangle is valid based on the angle measures?
  5. What happens to the angles if we change the position of point OO inside the triangle?

Tip: When solving geometry problems, always check if the sum of angles or sides satisfies the known properties of the shape (like the sum of angles in a triangle being 180180^\circ).

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Angle Sum in Triangles

Formulas

Sum of angles in a triangle = 180°

Theorems

Triangle Angle Sum Theorem

Suitable Grade Level

Grades 7-9