Question

in the triangle below, with right angle ∠k, suppose that m∠i=(4x−2)° and m∠j=(5x−34)°. find the degree measure of each angle in the triangle. m∠i = \\(\square\\)° m∠j = \\(\square\\)° m∠k = \\(\square\\)°

Explanation:

Step1: Recall triangle angle sum

In a triangle, the sum of angles is \(180^\circ\). Since \(\angle K\) is a right angle, \(m\angle K = 90^\circ\). So, \(m\angle I + m\angle J + m\angle K = 180^\circ\). Substituting the given expressions: \((4x - 2) + (5x - 34) + 90 = 180\).

Step2: Solve for \(x\)

Combine like terms: \(4x + 5x - 2 - 34 + 90 = 180\) → \(9x + 54 = 180\). Subtract 54: \(9x = 126\). Divide by 9: \(x = 14\).

Step3: Find \(m\angle I\)

Substitute \(x = 14\) into \(4x - 2\): \(4(14) - 2 = 56 - 2 = 54^\circ\).

Step4: Find \(m\angle J\)

Substitute \(x = 14\) into \(5x - 34\): \(5(14) - 34 = 70 - 34 = 36^\circ\).

Step5: Confirm \(m\angle K\)

Since it's a right angle, \(m\angle K = 90^\circ\).

Answer:

\(m\angle I = \boldsymbol{54}^\circ\)
\(m\angle J = \boldsymbol{36}^\circ\)
\(m\angle K = \boldsymbol{90}^\circ\)