10.4 Inscribed Angles and Polygons in Circles

​Inscribed Angles and Polygons in Circles

Geometry Standard: Geo.M.G.C.A.02

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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Learning Objective: (01/30/2023)

I can solve problems involving arcs and inscribed angles in circles.

Multiple Choice

RECALL: Arcs and Central Angles

What is the measure of angle EDC as shown in the figure?

• 35^o

• 215^o

• 75^o

• 70^o

Multiple Choice

RECALL: Arcs and Central Angles

What is the measure of major arc FHG as shown in the figure? Note that one full circle = 360^o

• 290^o

• 215^o

• 225^o

• 210^o

Inscribed Angles in Circles

Fill in the Blank

If angle ABC is inscribed in a circle then its measure is _____ of its intercepted arc AC.

Multiple Choice

If the inscribed angle measures 20^o, find the measure of the intercepted arc KL.

• $$30\degree$$

• $$80\degree$$

• $$10\degree$$

• $$40\degree$$

Multiple Choice

Find the measure of arc KML.

• $$45\degree$$

• $$180\degree$$

• $$100\degree$$

• $$90\degree$$

Multiple Choice

What are the measures of arc EG and angle DEF respectively?

• $$m\angle DEF=80\degree$$ $$arc\ EG=50\degree$$

• $$m\angle DEF=79\degree$$ $$arc\ EG=39\degree$$

• $$m\angle DEF=39\degree$$ $$arc\ EG=58\degree$$

• $$m\angle DEF=80\degree$$ $$arc\ EG=56\degree$$

Multiple Choice

Notice that both angles D and G intercept the same arc EH. Find the measure of angle x?

• x = 47

• x = 94

• x = 23.5

• x = 45

• Arc AC is a semicircle and measures 180^o

• Angle ABC is a right angle and measures 90^o

Inscribed Right Triangle

Multiple Choice

Arc MK is a semicircle, what is the measure of angle MLK?

• m∠MLK = 180^o

• m∠MLK = 50^o

• m∠MLK = 40^o

• m∠MLK = 90^o

Multiple Choice

Arc KM is a semicircle, what is the measure of angle LKM?

• m∠MLK = 180^o

• m∠MLK = 50^o

• m∠MLK = 40^o

• m∠MLK = 90^o

Inscribed Quadrilateral

Multiple Choice

Find the measure of each variable if the given quadrilateral is inscribed in the circle. Note that opposite angles add up to 180^o.

• x = 100^o

  y = 95^o

• x = 95^o

  y = 100^o

• x = 90^o

  y = 105^o

• x = 105^o

  y = 90^o

Multiple Choice

ABCD is inscribed in a circle as shown. Find the measure of each variable.

• x = 78^o

  y = 80^o

• x = 80^o

  y = 78^o

• x = 70^o

  y = 88^o

• x = 88^o

  y = 70^o

Summary

• The measure of an inscribed angle is one-half the measure of the intercepted arc.

• If two angles intercept the same arc, then the two angles are congruent.

• An inscribed right angle intercepts a semicircle or an arc which measures 180^o.

• The opposite angles of an inscribed quadrilateral add up to 180^o or are supplementary.

Multiple Choice

Use the inscribed angle theorem to find the value of angle x.

• x = 50^o

• x = 51^o

• x = 52^o

• x = 54^o

Multiple Choice

Look at the figure on the left, what is the measure of arc DC?

• 100^o

• 76^o

• 45^o

• 90^o

Multiple Choice

If angle D intercepts arc AB, then which angle is congruent to angle D?

• angle C

• angle A

• angle B

• cannot be determined

Multiple Choice

In the figure, arc BD is a semicircle, what must be the measure of angle A?

• angle A = 80^o

• angle A = 45^o

• angle A = 90^o

• angle A = 180^o

Multiple Choice

A quadrilateral is inscribed in a circle. Find the value of x and y. Hint: opposite angles add up to 180^o

• x = 100^o, y = 109^o

• x = 109^o, y = 100^o

• x = 99^o, y = 101^o

• x = 101^o, y = 99^o