Question

in the triangle below, with right angle $\angle m$, suppose that $m\angle k=(5x + 16)\degree$ and $m\angle l=(4x - 7)\degree$.
find the degree measure of each angle in the triangle.
$m\angle k = \square\degree$
$m\angle l = \square\degree$
$m\angle m = 90\degree$

Explanation:

Step1: Recall triangle angle sum

In a triangle, the sum of angles is \(180^\circ\). For right triangle \(KLM\) (right-angled at \(M\)), we have \(m\angle K + m\angle L + m\angle M = 180^\circ\). Substituting known values: \((5x + 16) + (4x - 7) + 90 = 180\).

Step2: Simplify and solve for \(x\)

Combine like terms: \(9x + 99 = 180\). Subtract 99: \(9x = 81\). Divide by 9: \(x = 9\).

Step3: Find \(m\angle K\)

Substitute \(x = 9\) into \(m\angle K = 5x + 16\): \(5(9) + 16 = 45 + 16 = 61\).

Step4: Find \(m\angle L\)

Substitute \(x = 9\) into \(m\angle L = 4x - 7\): \(4(9) - 7 = 36 - 7 = 29\).

Answer:

\(m\angle K = \boldsymbol{61}^\circ\)
\(m\angle L = \boldsymbol{29}^\circ\)
\(m\angle M = 90^\circ\)