https://file.aiquickdraw.com/mathbot/file/1725888704827kgk4j6eo.jpg

The problem states that \(\triangle LMN \cong \triangle PQR\), meaning these triangles are congruent. Congruent triangles have corresponding sides and angles that are equal.

Given:
- \(LM = PQ\), \(LN = PR\), and \(MN = QR\).
- We are tasked with finding \(x + y\).

### Step-by-Step Approach:

1. **Corresponding Angles:**
   Since the triangles are congruent, the angles will match:
   - \(\angle L = \angle P\)
   - \(\angle M = \angle Q\)
   - \(\angle N = \angle R\)

2. **Using the given angles:**
   - In \(\triangle LMN\), we have \(\angle L = 55^\circ\).
   - In \(\triangle PQR\), we have \(\angle R = 30^\circ\).
   - The angle at \(M\) is given as \((x + 7)^\circ\), and the angle at \(Q\) is given as \((y - 7)^\circ\).

3. **Sum of interior angles in a triangle:**
   The sum of the angles in any triangle is \(180^\circ\). Let's apply this to both triangles.

   - For \(\triangle LMN\):
     \[
     55^\circ + (x + 7)^\circ + \angle N = 180^\circ
     \]
     Simplifying:
     \[
     62^\circ + x + \angle N = 180^\circ
     \]
     \[
     x + \angle N = 118^\circ \quad \text{(1)}
     \]

   - For \(\triangle PQR\):
     \[
     (y - 7)^\circ + 30^\circ + \angle P = 180^\circ
     \]
     Simplifying:
     \[
     y + 23^\circ + \angle P = 180^\circ
     \]
     \[
     y + \angle P = 157^\circ \quad \text{(2)}
     \]

4. **Relating \(\angle N\) and \(\angle R\):**
   Since \(\triangle LMN \cong \triangle PQR\), \(\angle N = \angle R\), and \(\angle P = \angle L\). So, from (1) and (2), we can substitute \(\angle N = 30^\circ\) (from \(\angle R\)) and \(\angle P = 55^\circ\) (from \(\angle L\)) into the equations.

   - From \((x + \angle N = 118^\circ)\):
     \[
     x + 30^\circ = 118^\circ
     \]
     Solving for \(x\):
     \[
     x = 88^\circ
     \]

   - From \((y + \angle P = 157^\circ

Triangles ΔLMN and ΔPQR are congruent such that LM = PQ, LN = PR, and MN = QR. Find the value of x + y given that Angle L = 55°, Angle R = 30°, Angle M = (x+7)°, and Angle Q = (y-7)°.

Explore how to find the value of x + y in congruent triangles ΔLMN and ΔPQR using the properties of congruent triangles and corresponding angles. This problem involves understanding the congruence criteria and the sum of interior angles in a triangle. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation