Clock Activity 10.4• Please begin this activity

• In this activity, you will use the degrees of a clock to make an conjecture about inscribed angles.

Clock Activity 10.4

2 *Angle measure Created = Arc Measure

Inscribed angles whose endpoints are the diameter form right angles

2 *Angle measure Created = Arc Measure

10.4Use Inscribed Angles

and Polygons

Theorems

10-7: If an angle is inscribed in a circle, then the measure of the angle equals one-half

the measure of its intercepted arc.

10-8: If two inscribed angles of a circle or congruent circles intercept congruent arcs of the same arc, then the angles are congruent.

What do you think?

- If a second angle intercepted the same arc?

Theorems

10-9: If an inscribed angle of a circle intercepts a semicircle, then the angleis a right angle.

What do you think?

- If an angle intercepted a semicircle?

Theorems

10-10: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

What do you think?

- With an inscribed angle, what do you think the opposite angles would equal? Why?

Class Exercises1. Explain how an intercepted arc and an inscribed angle are

related.

2. ∆ABC is inscribed in a circle so that BC is a diameter. What type of triangle is ∆ ABC? Explain your answer.

3. In the circle at the right, mST = 68. Find the m<1 and m<2 .

R S

Q

TP

1

2

The inscribed angle measures half of the intercepted arc.

Since the inscribed angle intercepts in a semicircle, the angle is a right angle. Therefore, the triangle is a right triangle.

m<1 = 34

m<2 = 34

Class Exercises4. In circle A:

and PQ RS. Findthe measures ofangles 1,2,3 and 4.

A

Q R

P

T

S41 2

3

1161 xm1992 xm2543 ym934 ym

3y – 9 = 4y - 25

y = 166x + 11 + 9x + 19 = 90

15x + 30 = 90x = 4

m<1 = 35, m<2 = 55, m<3 = 39, and m<4 = 39

Clock Activity 10.5

• In this activity, you will use the degrees of a clock to make an conjecture about different types of angle relationships in circles using tangents, chords, and secants.

10.5Apply Other Angle

Relationships in Circles

Terms:

Secant:

Theorems:

If a _____________ and a _______________ intersect at the point oftangency, then the measure of each angle formed is _______________ the measure of its intercepted ______________.

If m<RTW = 50, then arc RT = _________If arc RT = 86, then m<RTW = _________ and m<RTV = ________.

A line that intersects a circle in exactly two places. It contains a chord of the circle.

secant tangentone-half

arc

10043

137

If two _________________ intersect in the ______________ of a circle, then

the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the ___________ and its ______________ angle. Similar to finding the average of the arc measures.

Formula: ______________________ If arc ST = 30 and arc RU = 80 then If arc ST = 30 and m<1 = 50,

m<3 = ______ and m<2 = _______ then arc RU = ______

secants interior

angle vertical

23

RUSTm

55 125 70

If two secants, a secant and a tangent, or two tangents intersect in the ___________ of a circle, then the measure of the angle formed is __________ the positive difference of the measures of the ___________________ arcs. Case 1: two secants Case 2: one secant / Case 3: 2 tangents

one tangent

exterior one-halfintercepted

Examples1. In Circle Q, m<CQD = 120, mBC = 30, and m<BEC = 25. Find each measure.

a. mDC b. mAD c. mAB d. m<QDC

120 80 130 30

2. Use Circle K to find the value of x. <R is formed by a secant and a tangent.

)5.2(2

50)54(

x

x x = 25

Examples3. Use Circle S to find the value of y.

282

)360(

yy y = 152

22°x°

85°

126.5°

Classwork • Finish ws