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

HC Is A Diameter HA 830 BC 50 HD 135 GF 32 HG 45 And FE 55 Findthe Measures Of Angle 1 class=

Sagot :

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

[tex]\hat{HC}=180\degree[/tex]

From the diagram, HC can be expressed as follows:

[tex]\hat{HC}=\hat{HA}+\hat{AB}+\hat{BC}[/tex]

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

[tex]\begin{gathered} 180\degree=83\degree+\hat{AB}+50\degree \\ 180\degree-83\degree-50\degree=\hat{AB} \\ 47\degree=\hat{AB} \end{gathered}[/tex]

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

[tex]m\angle1=\frac{1}{2}(\hat{HD}-\hat{AB})[/tex]

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

[tex]\begin{gathered} m\angle1=\frac{1}{2}(135\degree-47\degree) \\ m\angle1=44\degree \end{gathered}[/tex]

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