Angle Relationships Inside and Outside Circles

Angle Relationships Inside and Outside
Circles (Day 1)

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

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.

Vocabulary

If an angle is formed by two
intersecting chords or secants of a circle such that the vertex of the angle is in the interior of the circle, then the measure of the angle is half of the sum of the measures of the arcs
intercepted by the angle and its vertical angle.

The degree measure of a minor arc is equal to the degree measure of its central angle. The degree measure of a major arc is determined by subtracting the degree measure of the minor arc from 360°.

Adjacent arcs are two arcs of the same circle sharing a
common endpoint.

Vocabulary

The measure of an arc formed by two
adjacent arcs is the sum of the measures of the two arcs.

If a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a right triangle.

The measure of an inscribed angle is half the measure of its intercepted arc.

If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.

Degree measure of an Arc

Math Response

Given: minor arc AB = $$60\degree$$

Find the angle measure for major arc ABC.

Type answer here

Correct Answer: 300

Deg^°

Rad

Math Response

Given : major arc ACB = $$115\degree$$

Find: minor arc AB

Type answer here

Correct Answer: 145

Deg^°

Rad

Math Response

Given:

minor arc = 2x +8

major arc = $$232\degree$$

Find x

Type answer here

Correct Answer: 112

Deg^°

Rad

Math Response

Given:

minor arc ST = 3x+1

major arc SRT = 6x-1

Find x

Type answer here

Correct Answer: 40

Deg^°

Rad

Adjacent Arcs

Arc Addition Postulate

Math Response

Find the length of arc AC

Type answer here

Correct Answer: 29

Deg^°

Rad

Math Response

Find the length of arc AC

Type answer here

Correct Answer: 32

Deg^°

Rad

Math Response

Given: major arc ACB = 219

minor arc CB = 5

Find the value for x

Type answer here

Correct Answer: 7

Deg^°

Rad

Interior Angle of a Circle Theorem

Multiple Choice

Find the value for x

• $$100\degree$$

• 
  $$129\degree$$

• 
  $$71\degree$$

• $$200\degree$$

Multiple Choice

Find the value for x

• $$120\degree$$

• $$206\degree$$

• $$103\degree$$

• $$360\degree$$

Math Response

Find the value of x.

Type answer here

Correct Answer: 43

Deg^°

Rad

Inscribe Angle Theorem

Multiple Choice

Find the value for x

• $$15\degree$$

• $$30\degree$$

• $$45\degree$$

• $$60\degree$$

Multiple Choice

Find the value of x.

Click the all equivalent answers.

• $$34\degree$$

• $$\frac{17}{2}\degree$$

• $$26\degree$$

• $$8.5\degree$$

Math Response

Find the value of x.

Type answer here

Correct Answer: 16

Deg^°

Rad

Inscribed Right Triangle-Diameter

Math Response

Solve for x.

Type answer here

Correct Answer: 55

Deg^°

Rad

Math Response

Find the value of x

Type answer here

Correct Answer: 23

Deg^°

Rad

Multiple Choice

Given:

$$\Delta ABC$$ is a 30-60-90 $$\Delta$$

$$\angle A$$ = $$30\degree$$

$$\overline{AB}$$ = 12 m

Find the side length for $$\overline{CB}$$

• 12 m

• $$12\ \sqrt[]{3}$$ m

• $$6\ \sqrt[]{3}$$ m

• 6 m

Inscribe Quadrilateral-Opposite Angles Theorem

Math Response

Find the value of x

Type answer here

Correct Answer: 100

Deg^°

Rad

Multiple Choice

Solve for x and y

(Answer notation: x;y)

• $$72\degree;\ 92\degree$$

• $$252\degree\ ;\ 272\degree$$

• $$92\degree;\ 72\degree$$

• $$20\degree\ ;\ 196\degree$$

Open Ended

Find values of x, y, and z.

Answer notation:

x=

y=

z=

Exterior Angles of a Circle Theorem

​Exterior Angles of a Circle Theorem - If an angle is formed by two intersecting chords or secants of a circle such that the vertex of the angle is in the exterior of the circle, then the measure of the angle is half of the difference of the measures of the arcs intercepted by the angle.

Math Response

Find the exterior angle the figure

Type answer here

Correct Answer: 35

Deg^°

Rad

Math Response

Find the exterior angle for the figure.

Type answer here

Correct Answer: 60

Deg^°

Rad

Math Response

Find the measure of the missing arc.

Type answer here

Correct Answer: 120

Deg^°

Rad

Tangent to a Circle Theorem

​Tangent to a Circle Theorem - A line drawn tangent to a circle is perpendicular to a radius of the circle drawn to the point of tangency.

Skills Practice and DOL