Question

in the triangle below, with right angle ∠i, suppose that m∠j = (4x + 1)° and m∠k = (2x + 47)°. find the degree measure of each angle in the triangle. m∠i = □° m∠j = □° m∠k = □°

Explanation:

Step1: Recall triangle angle sum

In a right triangle, one angle is \(90^\circ\), and the sum of angles is \(180^\circ\). So \(m\angle I = 90^\circ\), and \(m\angle J + m\angle K + m\angle I = 180^\circ\). Substitute \(m\angle J=(4x + 1)^\circ\), \(m\angle K=(2x + 47)^\circ\), \(m\angle I = 90^\circ\):
\((4x + 1)+(2x + 47)+90 = 180\)

Step2: Solve for \(x\)

Combine like terms: \(6x + 138 = 180\).
Subtract 138: \(6x = 180 - 138 = 42\).
Divide by 6: \(x=\frac{42}{6}=7\).

Step3: Find \(m\angle J\)

Substitute \(x = 7\) into \(m\angle J=(4x + 1)^\circ\):
\(4(7)+1 = 28 + 1 = 29^\circ\).

Step4: Find \(m\angle K\)

Substitute \(x = 7\) into \(m\angle K=(2x + 47)^\circ\):
\(2(7)+47 = 14 + 47 = 61^\circ\).

Answer:

\(m\angle I = 90^\circ\), \(m\angle J = 29^\circ\), \(m\angle K = 61^\circ\)