10.3 Arcs and Chords & 10.4 Inscribed Angles

10.3 Arcs and Chords & 10.4 Inscribed Angles

Objectives • Recognize and use relationships between arcs, chords, and diameters • Find measures of inscribed angles • Find measures of angles of inscribed polygons

S R B A Arcs and Chords • Theorem 10.2:In a or  s, two minor arcs are  iff their corresponding chords are .

Diameters and Chords • Theorem 10.3:In a , if a diameter (or radius) is ┴ to a chord, then it bisects the chord and its arc. If JK ┴ LM, then MO  LO and arc LK  arc MK.

More about Chords • Theorem 10.4:In a or in  s, two chords are  iff they are equidistant from the center.

A C B Inscribed Angles • An inscribed angle is an angle that has its vertex on the circle and its sides are chords of the circle.

A C B Inscribed Angles • Theorem 10.5 (Inscribed Angle Theorem):The measure of an inscribed angle equals ½ the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). mACB = ½m or 2 mACB =

In and Find the measures of the numbered angles. Example 1:

First determine Example 1: Arc Addition Theorem Simplify. Subtract 168 from each side. Divide each side by 2.

So, m Example 1:

Answer: Example 1:

In and Find the measures of the numbered angles. Answer: Your Turn:

Inscribed Angles • Theorem 10.6:If two inscribed s intercept  arcs or the same arc, then the s are . mDAC  mCBD

Given: Prove: Example 2:

Proof: Statements Reasons 1. 1. Given 2. If 2 chords are , corr. minor arcs are . 2. 3. Definition of intercepted arc 3. 4. 4. Inscribed angles of arcs are . 5. 5. Right angles are congruent 6. 6. AAS Example 2:

Given: Prove: Your Turn:

Proof: Statements Reasons 1. 2. 3. 4. 5. 1. Given 2. Inscribed angles of arcs are . 3. Vertical angles are congruent. 4. Radii of a circle are congruent. 5. ASA Your Turn:

o Angles of Inscribed Polygons • Theorem 10.7:If an inscribed  intercepts a semicircle, then the  is a right . i.e. If AC is a diameter of , then the mABC = 90°.

A B O D C Angles of Inscribed Polygons • Theorem 10.8:If a quadrilateral is inscribed in a , then its opposite s are supplementary. i.e. Quadrilateral ABCD is inscribed in O, thus A and C are supplementary and B and D are supplementary.

ALGEBRATriangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and Example 3:

are right triangles. since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so . Example 3: Angle Sum Theorem Simplify. Subtract 105 from each side. Divide each side by 3.

Use the value of x to find the measures of Answer: Example 3: Given Given

ALGEBRATriangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and Answer: Your Turn:

Quadrilateral QRST is inscribed in If and find and Draw a sketch of this situation. Example 4:

To find we need to know To find first find Example 4: Inscribed Angle Theorem Sum of angles in circle=360 Subtract 174 from each side.

To find we need to know but first we must find Example 4: Inscribed Angle Theorem Substitution Divide each side by 2. Inscribed Angle Theorem

Answer: Example 4: Sum of angles in circle=360 Subtract 204 from each side. Inscribed Angle Theorem Divide each side by 2.

Quadrilateral BCDE is inscribed in If and find and Answer: Your Turn:

Assignment • GeometryPg. 540 #11 – 29 Pg. 549 #8 – 10, 13 – 16, 18, 22 - 25 • Pre-AP Geometry Pg. 540 #11 – 33 Pg. 549 #8 – 10, 13 – 30

10.3 Arcs and Chords & 10.4 Inscribed Angles