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Warm up…. Page 543 quiz #’s 1 – 6, 9, 10. 10-4 Inscribed Angles. Inscribed Angles. If an angle is inscribed in a circle, then the measure of the angle equals ½ the measure of the intercepted arc The measure of the intercepted arc is 2 times the measure of the inscribed angle. B. A. D. C.

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Warm up
Warm up…

  • Page 543 quiz

    #’s 1 – 6, 9, 10



Inscribed angles
Inscribed Angles

  • If an angle is inscribed in a circle, then the measure of the angle equals ½ the measure of the intercepted arc

  • The measure of the intercepted arc is 2 times the measure of the inscribed angle

B

A

D

C


Intercepting the same arc
Intercepting the same arc

  • If 2 inscribed angles intercept congruent arcs or the same arc then the angles are congruent

A

B

B

A

C

F

D

C

D

E


Angles of inscribed polygons
Angles of inscribed polygons

  • If an inscribed angle intercepts a semicircle, the angle is a right angle

A

D

B

C

ADC is a semicircle, so m<ABC = 90°


Example 1
Example 1:

  • mWX=20, mXY=40, mUZ=108 and mUW=mYZ. Find the measures of the numbered angles.

W

X

3

Y

4

5

F

2

U

Z

1

T


Example 2
Example 2:

  • Triangles TVU and TSU are inscribed in circle P, m<2 = x+9 and m<4 =2x+6,

  • Find the measure of the numbered angles

U

V

3

4

S

P

1

2

T


Inscribed quadrilaterals
Inscribed Quadrilaterals

  • If a quadrilateral is inscribed in a circle then its opposite angles are supplementary



Secants
Secants

  • A line that intersects a circle in exactly 2 points

  • When 2 secant lines intersect in the interior of a circle the measure of an angle formed is ½ the sum of the measures of the intercepted arcs and its vertical angle

A

D

m<1 = ½(mAC + mBD)

m<2 = ½(mAD + mBC)

2

1

B

C


Secant and tangent
Secant and Tangent

  • If a secant and a tangent intersect at a point of tangency, then the measure of each angle formed is ½ the measure of the intercepted arc

  • Example: Find m<RPS if mPT =114 and mTS = 136

R

P

Q

S

114°

T

136°


Intersection outside a circle
Intersection outside a circle

D

  • Two secants – m<A = ½(mDE – mBC)

  • Secant-tangent – m<A=1/2(mDC – mBC)

  • Two tangents – m<A=1/2(mBDC – mBC)

B

A

C

E

D

B

A

C

B

D

A

C


Examples find x
Examples: Find x

40°

6x

55°

141°

x

62°


Assignment
Assignment:

  • Page 549 #13-16 all

  • Page 564 # 13-27 odd


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