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