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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

Inscribed AnglesTangents

Notes 27 – Sections 10.4 & 10.5

essential learnings
Essential Learnings
  • Students will understand and be able to find measures of inscribed angles and intercepted arcs.
vocabulary
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
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
Example 1
  • Find each measure.
theorem 10 7
Theorem 10.7
  • If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.
example 2
Example 2
  • Find the measure of ∠R.
theorem 10 8
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
Example 3
  • Find the measure of ∠B.
theorem 10 9
Theorem 10.9
  • If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
example 4
Example 4
  • Find the measures of angles A and D.
tangents
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

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

tangency theorem
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
Example 5

Determine whether XY is tangent to the circle.

tangent segments conjecture
Tangent Segments Conjecture

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

example 6
Example 6

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

circumscribed polygons
Circumscribed Polygons

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

example 7
Example 7

Find the value of x. Then find the perimeter.

assignments
Assignments

p.713: 11 – 20, 23 – 30, 36

p. 722: 13 – 22, 24 – 27, 30

Unit Study Guide 9