Question

angle relationships

1. find m∠dce.
2. find m∠hki.
3. find the value of x.

4)
∠deg = 85°
∠gef =
∠def = 142°

Explanation:

Step1: Find \(m\angle DCE\) in the first - figure

The sum of angles around a point is \(360^{\circ}\). In the first figure, if we assume the right - angle is \(90^{\circ}\), then \(m\angle DCE=360^{\circ}-90^{\circ}-67^{\circ}=203^{\circ}\).

Step2: Find \(m\angle HKI\) in the second - figure

The sum of angles around a point is \(360^{\circ}\). So \(m\angle HKI = 360^{\circ}-127^{\circ}-151^{\circ}=82^{\circ}\).

Step3: Find the value of \(x\) in the third - figure

The sum of angles around a point is \(360^{\circ}\), and we have two right - angles (\(90^{\circ}\) each). So \((3x - 1)+4x+90 + 90=360\). Combine like terms: \(7x+180 = 360\). Subtract 180 from both sides: \(7x=360 - 180=180\). Then \(x=\frac{180}{7}\approx25.71\).

Step4: Find \(\angle GEF\) in the fourth - figure

We know that \(\angle DEF = 142^{\circ}\) and \(\angle DEG = 85^{\circ}\). Then \(\angle GEF=\angle DEF-\angle DEG=142^{\circ}-85^{\circ}=57^{\circ}\).

Answer:

1. \(m\angle DCE = 203^{\circ}\)
2. \(m\angle HKI = 82^{\circ}\)
3. \(x=\frac{180}{7}\)
4. \(\angle GEF = 57^{\circ}\)