Answer:
6. ∠GHJ ≅ ∠GIJ = 73°
10. ∠Q = 89°, ∠R = 123°, ∠S = 91°
Step-by-step explanation:
The relations applicable to these problems are ...
- the measure of an inscribed angle is half the measure of the intercepted arc
- the sum of the arc measures in a circle is 360°
Opposite angles of an inscribed quadrilateral intercept arcs that total 360°, so those opposite angles must total 360°/2 = 180°.
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6.
The missing arc measure GJ is ...
68° +31° +115° +GJ = 360°
GJ = 360° -214° = 146°
Both of inscribed angles GHJ and GIJ intercept arc GJ, so both will have measure ...
∠GHJ ≅ ∠GIJ = 146°/2 = 73°
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8.
∠Q intercepts arc PSR, which is shown as ...
arc PSR = arc PS +arc SR
arc PSR = 137° +41° = 178°
The measure of angle Q is half this:
∠Q = (1/2)×arc PSR = 178°/2 = 89°
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∠R is opposite ∠P in the inscribed quadrilateral, so is supplementary to it:
∠R = 180° -57° = 123°
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∠S is opposite ∠Q in the inscribed quadrilateral, so is supplementary to it:
∠S = 180° -89° = 91°
