Question

1. find the value of x. then find the measure of each labeled angle.

a -----> b
| x°
|
d ┐ (x - 58)°
-----> c
m∠a = 90°, m∠b = 119°,
m∠c = 61°, m∠d = 90°

Explanation:

Step1: Recall the sum of angles in a quadrilateral

The sum of the interior angles of a quadrilateral is \(360^\circ\). So we have the equation: \(m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ\). But we can also use the given expressions for \(\angle B\) and \(\angle C\) to set up an equation in terms of \(x\). We know \(\angle B = x^\circ\) and \(\angle C=(x - 58)^\circ\), \(\angle A = 90^\circ\), \(\angle D = 90^\circ\). So the equation is \(90 + x+(x - 58)+90 = 360\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(90+90 - 58+x+x=360\). Calculate \(90 + 90-58=122\), and \(x + x = 2x\). So the equation becomes \(122+2x=360\).

Step3: Solve for \(x\)

Subtract 122 from both sides: \(2x=360 - 122\). \(360-122 = 238\), so \(2x = 238\). Divide both sides by 2: \(x=\frac{238}{2}=119\).

Step4: Find the measure of \(\angle C\)

Since \(\angle C=(x - 58)^\circ\), substitute \(x = 119\) into the expression: \(119-58 = 61^\circ\). And we can verify that \(90+119 + 61+90=360\) (because \(90+119=209\), \(61 + 90 = 151\), and \(209+151 = 360\)).

Answer:

The value of \(x\) is \(119\). The measure of \(\angle B\) is \(119^\circ\) and the measure of \(\angle C\) is \(61^\circ\).