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

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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.

Draw

1.) Draw secants KL and MN.

2.) Where do the secants intersect?

3. 3.) Label this point as point P on the figure.

4.) Draw chord KN.

Open Ended

Can you determine the measure of ∠KPM with the information you have so far? If so, how? Explain your reasoning

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°

Multiple Choice

How does ∠KTN relate to △KTN?

• ∠KTN is an interior angle of △KTN.

• ∠KTN is an exterior angle of △KTN.

Multiple Choice

What is the measure of ∠KTN?

• 40°

• 15°

• 25°

• 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.

Multiple Choice

Determine measurement of arc FI.

GIVEN: m∠K = 20°
GIVEN: measure of arc GJ = 80°

• 100°

• 120°

• 80°

• 60°

Multiple Choice

Determine m∠X.

GIVEN: measure of arc VW = 40°
GIVEN: measure of arc TU = 85°

• 20°

• 125°

• 22.5°

• 42.5°

Multiple Choice

Determine m∠D.

GIVEN: measure of arc ZXC = 120°
GIVEN: measure of arc CB = 30°

• 80°

• 150°

• 90°

• 115°

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.

Fill in the Blanks

DOL & Skills Check!