Question

HC is a diameter. HA = 830, BC = 50°, HD = 135, GF = 32°, HG = 45°, and FE = 55°. Findthe measures of angle 1

Answer 1

Given that HC is the diameter of the circle, then:

`\hat{HC}=180\degree`

From the diagram, HC can be expressed as follows:

`\hat{HC}=\hat{HA}+\hat{AB}+\hat{BC}`

Substituting with HC = 180°, HA = 83°, and BC = 50°, and solving for AB:

`\begin{gathered} 180\degree=83\degree+\hat{AB}+50\degree \\ 180\degree-83\degree-50\degree=\hat{AB} \\ 47\degree=\hat{AB} \end{gathered}`

The relation between the angle outside the circle, ∠1, and the intersected arcs AB and HD are:

`m\angle1=(1)/(2)(\hat{HD}-\hat{AB})`

Substituting with HD = 135°, and AB = 47°, we get:

`\begin{gathered} m\angle1=(1)/(2)(135\degree-47\degree) \\ m\angle1=44\degree \end{gathered}`