10.3 Arcs and Chords & 10.3 Arcs and Chords & 10.4 Inscribed Angles10.4 Inscribed Angles

ObjectivesObjectives

Recognize and use relationships Recognize and use relationships between arcs, chords, and between arcs, chords, and diametersdiameters

Find measures of inscribed anglesFind measures of inscribed angles

Find measures of angles of Find measures of angles of inscribed polygonsinscribed polygons

Arcs and ChordsArcs and Chords

Theorem 10.2:Theorem 10.2: In a or In a or s, s, two minor arcs are two minor arcs are iff their iff their corresponding chords are corresponding chords are ..

B

R

A

S

Diameters and ChordsDiameters and Chords

Theorem 10.3:Theorem 10.3: In a , if a In a , if a diameter (or radius) is ┴ to a diameter (or radius) is ┴ to a chord, then it bisects the chord chord, then it bisects the chord and its arc. and its arc.

If JK ┴ LM, then MO LO and arc LK arc MK.

More about ChordsMore about Chords

Theorem 10.4:Theorem 10.4: In a or in In a or in s, two chords are s, two chords are iff they are iff they are equidistant from the center. equidistant from the center.

Inscribed AnglesInscribed Angles

An An inscribed angleinscribed angle is an angle is an angle that has its vertex on the circle that has its vertex on the circle and its sides are chords of the and its sides are chords of the circle.circle.

C

A

B

Inscribed AnglesInscribed Angles

Theorem 10.5 Theorem 10.5 (Inscribed Angle (Inscribed Angle Theorem)Theorem)::The measure of an The measure of an inscribed angle inscribed angle equals ½ the equals ½ the measure of its measure of its intercepted arc (or intercepted arc (or the measure of the the measure of the intercepted arc is intercepted arc is twice the measure of twice the measure of the inscribed angle).the inscribed angle).

C

A

B

mACB = ½m or

2 mACB =

In and Find the measures of the numbered angles.

Example 1:Example 1:

Arc Addition Theorem

Simplify.

Subtract 168 from each side.

Divide each side by 2.

First determine

Example 1:Example 1:

So, m

Example 1:Example 1:

Answer:

Example 1:Example 1:

In and Find the measures of the numbered angles.

Answer:

Your Turn:Your Turn:

Inscribed AnglesInscribed Angles

Theorem 10.6:Theorem 10.6:If two inscribed If two inscribed s intercept s intercept arcs or arcs or the same arc, then the the same arc, then the s are s are . .

mDAC mCBD

Given:

Prove:

Example 2:Example 2:

Proof: Statements Reasons

1. Given1.

2. 2. If 2 chords are , corr. minor arcs are .

3. 3. Definition of intercepted arc

4. 4. Inscribed angles of arcs are .

5. 5. Right angles are congruent

6. 6. AAS

Example 2:Example 2:

Prove:

Given:

Your Turn:Your Turn:

1. Given

2. Inscribed angles of arcs are .

3. Vertical angles are congruent.

4. Radii of a circle are congruent.

5. ASA

Proof: Statements Reasons

1.

2.

3.

4.

5.

Your Turn:Your Turn:

Angles of Inscribed Angles of Inscribed PolygonsPolygons

Theorem 10.7:Theorem 10.7:If an inscribed If an inscribed intercepts a intercepts a semicircle, then the semicircle, then the is a right is a right ..

i.e.i.e. If AC is a If AC is a diameter of , then diameter of , then the mthe mABC = 90ABC = 90°.°.

o

Angles of Inscribed Angles of Inscribed PolygonsPolygons

Theorem 10.8:Theorem 10.8:If a quadrilateral is If a quadrilateral is inscribed in a , then inscribed in a , then its opposite its opposite s are s are supplementary.supplementary.

i.e.i.e. Quadrilateral Quadrilateral ABCD ABCD is inscribed in O, is inscribed in O, thus thus A and A and C are C are supplementary and supplementary and B and B and D are D are supplementary. supplementary.

D

A

C

BO

ALGEBRA Triangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and

Example 3:Example 3:

are right triangles. since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so .

Angle Sum Theorem

Simplify.

Subtract 105 from each side.

Divide each side by 3.

Example 3:Example 3:

Use the value of x to find the measures of

Given Given

Answer:

Example 3:Example 3:

Answer:

ALGEBRA Triangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and

Your Turn:Your Turn:

Quadrilateral QRST is inscribed in If and find and

Draw a sketch of this situation.

Example 4:Example 4:

To find we need to know

To find first find

Inscribed Angle Theorem

Sum of angles in circle = 360

Subtract 174 from each side.

Example 4:Example 4:

Inscribed Angle Theorem

Substitution

Divide each side by 2.

To find we need to know but first we must find

Inscribed Angle Theorem

Example 4:Example 4:

Sum of angles in circle = 360

Subtract 204 from each side.

Inscribed Angle Theorem

Divide each side by 2.

Answer:

Example 4:Example 4:

Answer:

Quadrilateral BCDE is inscribed in If and find and

Your Turn:Your Turn:

AssignmentAssignment

GeometryGeometryPg. 540 #11 – 29Pg. 540 #11 – 29Pg. 549 #8 – 10, 13 – 16, 18, 22 Pg. 549 #8 – 10, 13 – 16, 18, 22

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Pre-AP Geometry Pre-AP Geometry Pg. 540 #11 – 33 Pg. 540 #11 – 33 Pg. 549 #8 – 10, 13 – 30 Pg. 549 #8 – 10, 13 – 30