Question

question
in the diagram below, \\( \overline { a b } \parallel \overline { c d } \\), \\( \overline { a d } \parallel \overline { b c } \\), \\( \overline { d e } \perp \overline { f c } \\), \\( \mathrm { m } \angle a d g = 54 ^ { \circ } \\) and \\( \mathrm { m } \angle d c g = 36 ^ { \circ } \\). find \\( \mathrm { m } \angle b \\).

you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

answer attempt 1 out of 5
\\( \mathrm { m } \angle b = \\) submit answer

Explanation:

Step1: Find \(m\angle DCF\)

Since \(DE\perp FC\), in right - triangle \(DGC\), \(m\angle DGC = 90^{\circ}\). Using the angle - sum property of a triangle (\(m\angle DGC+m\angle DCG + m\angle CDG=180^{\circ}\)), and \(m\angle DCG = 36^{\circ}\), we can find \(m\angle CDG\).

$$m\angle CDG=180^{\circ}-90^{\circ}-36^{\circ}=54^{\circ}$$

Step2: Find \(m\angle ADC\)

Given \(m\angle ADG = 54^{\circ}\), then \(m\angle ADC=m\angle ADG + m\angle CDG\)

$$m\angle ADC=54^{\circ}+54^{\circ}=108^{\circ}$$

Step3: Use the property of a parallelogram

Since \(AB\parallel CD\) and \(AD\parallel BC\), \(ABCD\) is a parallelogram. In a parallelogram, \(m\angle B+m\angle ADC = 180^{\circ}\) (adjacent angles of a parallelogram are supplementary)

$$m\angle B=180^{\circ}-m\angle ADC$$

Substitute \(m\angle ADC = 108^{\circ}\) into the formula:

$$m\angle B=180^{\circ}-108^{\circ}=72^{\circ}$$

Answer:

\(72\)