Inscribed Angles Tangents

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Inscribed Angles Tangents. Notes 27 – Sections 10.4 & 10.5. Essential Learnings. Students will understand and be able to find measures of inscribed angles and intercepted arcs. Vocabulary. Inscribed angle – an angle with the vertex on a circle and sides are chords in a circle.

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## Inscribed Angles Tangents

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### Inscribed AnglesTangents

Notes 27 – Sections 10.4 & 10.5

Essential Learnings
• Students will understand and be able to find measures of inscribed angles and intercepted arcs.
Vocabulary
• Inscribed angle – an angle with the vertex on a circle and sides are chords in a circle.
• Intercepted arc – an arc with endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle.

Intercepted arc

Inscribed angle

Inscribed Angle Theorem
• If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc.
Example 1
• Find each measure.
Theorem 10.7
• If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.
Example 2
• Find the measure of ∠R.
Theorem 10.8
• An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle.
Example 3
• Find the measure of ∠B.
Theorem 10.9
• If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Example 4
• Find the measures of angles A and D.
Tangents

Tangent – a line in the same plane as a circle that intersects the circle in exactlyone point called the point of tangency.

Common Tangent

Common tangent – a line, ray, or segment that is tangent to two circles in the same plane.

Tangency Theorem

In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

Example 5

Determine whether XY is tangent to the circle.

Tangent Segments Conjecture

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Example 6

Find x. Assume that segments that appear to be tangent are tangent. Round to the nearest tenth.

Circumscribed Polygons

A polygon is circumscribed about a circle if every side of the polygon is tangent to the circle.

Example 7

Find the value of x. Then find the perimeter.

Assignments

p.713: 11 – 20, 23 – 30, 36

p. 722: 13 – 22, 24 – 27, 30

Unit Study Guide 9