Question

May I please receive help?

Answer 1

`\displaystyle m\angle L = 52^\circ, m\angle M = 52^\circ, \text{ and } m\angle N = 76^\circ`

Explanation:

Recall that the interior angles of a triangle must total 180°. In other words:

`\displaystyle m\angle L + m\angle N + m\angle M = 180^\circ`

Since LN ≅ MN, ∠L ≅ ∠M. Hence:

`\displaystyle \begin{aligned} m\angle L + m\angle N + (m\angle L ) & = 180^\circ \\ \\ 2m\angle L + m\angle N & = 180^\circ \end{aligned}`

Substitute and solve for x:

`\displaystyle \begin{aligned} 2(5x-13)+ (4x+24) & = 180\\ \\ (10x -26) + (4x +24) & = 180 \\ \\ 14x - 2 & = 180 \\ \\ x & = (182)/(14) =13 \end{aligned}`

Therefore, the measure of ∠N is:

`\displaystyle \begin{aligned} m\angle N & = (4x+24)^\circ \\ \\ & =(4(13)+24)^\circ \\ \\ & = 76^\circ\end{aligned}`

And the measures of ∠L and ∠N are:

`\displaystyle \begin{aligned} m\angle N = m\angle L & = (5x-13)^\circ \\ \\ & = (5(13)-13)^\circ \\\ \\ & = 52^\circ \end{aligned}`

In conclusion:

`\displaystyle m\angle L = 52^\circ, m\angle M = 52^\circ, \text{ and } m\angle N = 76^\circ`