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Tomtit [17]
3 years ago
8

Angle measures and segment lengths. Algebra Find the value of each variable.

Mathematics
1 answer:
Katyanochek1 [597]3 years ago
6 0

(9)

<em>x</em> ° = 1/2 (130° - 30°) = 50°

(due to a theorem about intersecting secants/tangents)

(11) The labeled angle subtends a minor arc of meaure 120°, which means the larger arc has a measure of 360° - 120° = 240°. Then

<em>x</em> ° = 1/2 (240° - 120°) = 60°

(due to the same theorem)

(13) The labeling here is a bit confusing. I'm not sure what the 70° is referring to. It occurs to me that it might be info from a different exercise, so that <em>y </em>° is the measure of the angle made by the tangent to the circle with a vertex of the pentagon, and <em>x</em> ° is the measure of each arc that passes over an edge of the pentagon.

Each arc makes up 1/5 of the circle's circumferece, so

<em>x</em> ° = 360°/5 = 72°

The pentagon is regular, so each of its interior angles have the same measure of 108°. (Why 108°? Each exterior angle measures 360°/5 = 72°, since the exterior angles sum to 360°. Interior and exterior angles are supplementary, so the interior angles measure 180° - 72° = 108° each.)

The angles formed by the tangent to the circle are supplementary, so that

<em>y</em> ° + 108° + <em>y</em> ° = 180°

2<em>y</em> ° = 72°

<em>y</em> ° = 36°

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