Question

find ( mangle lmn ).

Explanation:

Step1: Use the property of corresponding angles

Since \( QL\parallel PO\), then \(16x + 15=10x + 9+180\) (exterior - angle property: the exterior angle of a triangle formed by a transversal and two parallel lines is equal to the sum of the non - adjacent interior angles).

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Step2: Calculate \(m\angle LMN\)

Substitute \(x = 29\) into \(16x+15\).

$$16\times29+15=464 + 15=479$$

(This is wrong, actually, we should use the property that \(QL\parallel PO\), and \(\angle LMN\) and \((10x + 9)^{\circ}\) are supplementary (same - side interior angles). So \(m\angle LMN=180-(10x + 9)\). First, find \(x\) from \(16x+15+10x + 9=180\) (linear pair or same - side interior angles sum to \(180^{\circ}\)).

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Then \(m\angle LMN=16x+15\), substitute \(x = 6\)

$$m\angle LMN=16\times6+15=96 + 15=111^{\circ}$$

Answer:

\(111^{\circ}\)