S2.T2.L7 - Angle Relationships Inside & Outside Circles

LO: SWBAT apply theorems about circles by determining the
measure of arcs, central angles, and inscribed angles, and prove the inscribed right triangle-diameter theorem, inscribed
quadrilateral-opposite angles theorem and interior angles of a circle theorem.

DOL: Given 5 problems, students will correctly apply theorems about circles by determining the measure of arcs, central angles and inscribed angles, and prove the inscribed right triangle-diameter theorem, inscribed quadrilateral -opposite angles theorem and interior angles of a circle theorem in at least 4 of 5 problems.

Geom.12A Apply theorems about circles, including relationships among angles, radii, chords, tangents and secants to solve
non-contextual problems.

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Lesson Vocabulary

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Exterior Angles of a Circle Theorem

Example

Tangent to a Circle Theorem

Example

The measure of an exterior angle of a circle is
equal to half the difference of the measures of
the arcs intercepted by the angle.

The tangent makes a right angle at the point of
tangency with the radius of a circle

Lesson Vocabulary

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Arc Addition Postulate

Example

(Review)

Inscribed Angle Theorem

Example

(Review)

Inscribed Quadrilateral-Opposite
Angles Theorem

Example

(Review)

Inscribed Right Triangle-Diameter
Theorem

Example

(Review)

Lesson Vocabulary

Answer the

essential question
on a response card.

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Interior Angles of
a Circle Theorem

Example

(Review)

degree measure of an
arc

Example

(Review)

adjacent arcs

Example

(Review)

1.Construct tangent lines to a circle through a point outside of the circle.

Activity 5.4: Exterior Angles of a Circle Theorem

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Step 1

Draw a circle with center point C and locate point P outside of the circle.

Step 2

Draw PC

Step 3

Construct the perpendicular bisector of line segment PC.

Step 4

Label the midpoint of the perpendicular bisector of line segment PC point M.

Step 5

Adjust the radius of your compass to the distance from point M to point C.

Step 6

Place the compass point on point M, and cut two arcs that intersect circle C.

Step 7

Label the two points at which the arcs cut through circle C point A and point B.

Step 8

Connect point P and A to form tangent line PA and connect point P and B to
form tangent line PB.

Line PA and line PB are
tangent to circle C.

2. Circle T with points K, L, M, and N is shown.

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c. Use the information given in the figure to name

the measures of any angles that you do know.
Explain how you determined your answers.

15°

30°

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Case 1: Exterior angle formed by a secant and a tangent

If a secant and a tangent intersect at the point of
tangency, then the measure of each angle formed
is half the measure of its intercepted arc.

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Case 2: Exterior angle formed by two secants

If two secants, a secant and a tangent, or two tangents intersect in the exterior
of a circle, then the measure of the angle formed is half the positive difference
of the measures of the intercepts arcs.

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Case 3: Exterior angle formed by two tangents

If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is half the positive difference of the measures of the intercepts arcs.

5.4 Section Recap

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What did you notice about the exterior angles of a circle
created by secants and tangents?

Sample Answer: It appears that the measure of an exterior angle of a
circle is equal to half the difference of the measures of the arcs
intercepted by the angle.

1.

Consider ∠UTV with vertex located on circle C . Line VW is drawn
tangent to circle C at point T.

Activity 5.5: Tangent to a Circle Theorem

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a. Determine and . Explain your

reasoning.

1.

Consider ∠UTV with vertex located on circle C . Line VW is drawn tangent
to circle C at point T.

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b. Determine m∠UTV and m∠UTW.

Explain your reasoning.

m∠UTV = 90° and m∠UTW = 90°
because the measure of an
inscribed angle is equal to half the
measure of its intercepted arc.