Download Lesson 10.4 Other Angle Relationships in Circles

Lesson 10.4
Other Angle Relationships in Circles
by Mrs. C. Henry
Objectives:
Goal 1: Use angles formed by tangents and chords
to solve problems in geometry.
Goal 2: Identify angles formed by lines that
intersect a circle to solve problems.
Goal 3: Utilize properties of tangent angles and
chords to solve real-life problems, such as finding
from how far away you can see fireworks.
10.4 Using Tangents and Chords
You know that the measure of an angle inscribed in a circle
is half the measure of its intercepted arc. This is true even if
one side of the angle is tangent to the circle.
Thm 10.12 Tangent Chord Angle
m 1 = ½ mAB
m 2 = ½ mBCA
10.4 Using Tangents and Chords
Using the Postulates and Theorems
Tangent Chord Angle Thm: Now it is your turn!
Work the following problem. Sketch the diagram.
Line m is tangent to the circle.
Find the measure of the red angle or arc.
10.4 Using Tangents and Chords
Using the Postulates and Theorems
Tangent Chord Angle Thm: Now it is your turn!
Work the following problem. Sketch the diagram.
Line m is tangent to the circle.
Find the measure of the red angle or arc.
m 1 = ½ (150°) = 75°
10.4 Using Tangents and Chords
If two lines intersect a circle, there are three places where
the lines can intersect.
2
on the circle
inside the circle
outside the circle
You know how to find angle and arc measures when lines
intersect on the circle. Now let’s look at an example of
inside the circle.
Thm 10.13 Crossed Chords
m 1 = ½ (mCD + mAB)
m 2 = ½ (mBC + mAD)
10.4 Using Tangents and Chords
Using the Postulates and Theorems
Crossed Chords Thm: Now it is your turn!
Work the following problem. Sketch the diagram.
10.4 Using Tangents and Chords
Using the Postulates and Theorems
Crossed Chords Thm: Now it is your turn!
Work the following problem. Sketch the diagram.
m RHW = ½ (mRW + mST)
m RHW = ½ (110° + 40°)
m RHW = ½ (150°)
m RHW = 75°
10.4 Using Tangents and Chords
Finally, let’s look at how to handle angle and arc measures
when the lines intersect outside the circle. Do you
remember what a secant is?
Thm 10.14
Exterior Intersecting
Tangents/Secants
A tangent and a secant, 2 tangents, or 2
secants intersect in the exterior of a circle.
m 1 = ½ (mBC – mAC)
m 2 = ½ (mPQR – mPR)
m 3 = ½ (mXY – mWZ)
Secant line (or segment): a line (or segment) that intersects a circle at two points.
10.4 Using Tangents and Chords
Using the Postulates and Theorems
Exterior Intersecting Tangents & Secants Thm:
Now it is your turn! Work the following problem. Sketch the
diagram.
A
10.4 Using Tangents and Chords
Using the Postulates and Theorems
Exterior Intersecting Tangents & Secants Thm:
Now it is your turn! Work the following problem. Sketch the
diagram.
A
m RST = ½ (mRAT – mRT)
m RST = ½ ((360° – 80°) – 80°)
m RST = ½ (280° – 80°)
m RST = ½ (200°)
m RST = 100°