Key Concepts
- Use volume of a prism.
- Find the volume of an oblique cylinder.
- Solve a real-world problem.
Theorem 10.11
If the tangent and a chord intersect at a point on a circle, then the measure of each angle
formed is one half the measure of its intercepted arc.
![Theorem 10.11](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2768.png)
![Theorem 10.11](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2769.png)
This proof will consist of two parts. The first and second equations of the theorem will be proved one at a time.
![This proof will consist of two parts. The first and second equations of the theorem will be proved one at a time.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2770.png)
Consider a diameter . Since is tangent to the circle at A, by the Tangent to Circle Theorem, and are perpendicular.
i.e ∠FAB + ∠BAD = 90° ………..Theorem 10.1
∠FAB =90° – ∠BAD
1/2m FB = 90° – ∠BAD……. Theorem 10.7
1/2m FB = 90° – m∠1
mFB = 2(90° – m∠1)
mFB = 180° – 2m∠1 ……………………….. 1
mFB +mAB =180° …………Arc addition postulate
180° – 2m∠1 + mAB = 180° …………………Substitute first equation
–2m∠1+ mAB =0
mFB +mAB =180° ………. Arc addition postulate
180° – 2m∠1 + mAB = 180° ……….. Substitute first equation
– 2m∠1 + mAB =0
– 2m∠1 = – mAB
2m∠1 = mAB
m∠1 = mAB (By solving equation)
Similarly, m∠2 = 1/2m AB
Example 1:
Line m is tangent to the circle. Find the measure of the red angle or arc
![Example 1:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2771.png)
Solution:
Theorem10.11: One half the measure of the intercepted arc.
m∠1 = SMALL ARC / 2
= 1/2 (1300)
= 65
∴ m∠1 = 650
Line M is tangent to the circle. Find the measure of the red angle or arc.
![Line M is tangent to the circle. Find the measure of the red angle or arc.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2772.png)
Solution:
Theorem10.11: One half the measure of the intercepted arc
m KJL/2= 1250 …………..
m KJL = 2.(1250)
= 2500
∴ m KJL= 2500
Guided practice for Example 1:
![Guided practice for example 1:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2773.png)
1. m∠1 2. mRST 3. mXY
1. m∠1
Solution:
Theorem10.11: One half the measure of the intercepted arc.
m∠1 = 1/2 (2100) …….
= 1050
∴ m∠1 = 1050
2. mRST
![mRST](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2774.png)
Solution:
Theorem10.11: One half the measure of the intercepted arc.
mRST / 2=980…………
m = 2(980)
= 1960
∴ mRST = 1960
3. mXY
![mXY](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2887.png)
Solution:
mXY/2=800
m = 2(800)
= 1600
∴ mXY = 1600
Intersecting Lines and Circles
If two lines intersect a circle, there are three places where the lines can intersect.
- On the circle
![On the circle](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2775.png)
- Inside the circle
![Inside the circle](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2776.png)
- Outside the circle
![Outside the circle](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2777.png)
- Angle inside the circle theorem
Theorem 10.12
If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
![Theorem 10.12](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/12-4.png)
Example 2:
Find the value of x.
![Example 2: Find the value of x.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2891.png)
Solution:
x° = 1/2 (mJM + mKL ) (Use Theorem 10.12)
x° = 1/2 (130°+ 156°) (Substitute)
x° = 1/2 (156°)
x° = 143° (Simplify)
Theorem 10.13
If a tangent and a secant two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
![Theorem 10.13](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2778.png)
![Theorem](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2779.png)
Example 3:
- Find the value of x.
![Find the value of x.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2780.png)
Solution:
The tangent CD and the secant CB intersect outside the circle.
x° = 1/2 ( 247° – 113°)
m ∠BCD = 1/2 (mAD – mBD) (Use theorem 10.13)
𝑥 ° = 1/2 ( 134° – 76°) (Substitute)
∴ 𝑥 = 51° (Simplify)
- Find the value of x
![Find the value of x](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2787.png)
Solution:
One half the measure of the difference of the intercepted arcs.
x° = 1/2 ( 247° – 113°)
𝑥 ° = 1/2 ( 134° ) (Substitute)
∴ 𝑥 = 67°
- Find the value of a.
![Find the value of a.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2788.png)
Solution:
![Solution:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2789.png)
One half the measure of the difference of the intercepted arcs
30 = 1/ 2(a−44)
60 = a−44
a = 104°
Example 4:
The Northern lights are bright flashes of colored light 50 and 200 miles above Earth.
Suppose a flash occurs 150 miles above Earth. What is the measure of arc BD, the portion
Of Earth from which the flash is visible?
(Earth’s Radis is approximately 4000 miles.)
![Example 4: solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2382.png)
Let’s Check your Knowledge
Find the value of x. Any lines that appear to be tangent can be assumed to be tangent.
![1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2793.png)
2
![2](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2794.png)
3.
![3](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2796.png)
4.
![4](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2798.png)
5.
![5](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2800.png)
6.
![6](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2801.png)
Answers
![1234](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2383.png)
![5](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2384.png)
![6](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2385.png)
Exercise
- Line m is tangent to the circle. Find the indicated measure.
i. m∠1
![i. m∠1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2809.png)
- m∠RST
![mRST](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2810.png)
- Find the value of x.
![2. Find the value of x.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2811.png)
- Find the value of x.
![Find the value of x.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2812.png)
- Find the value of the variable.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2813.png)
- Find the value of x.
![Find the value of x.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2814.png)
Concept Map
![Concept Map:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2808.png)
What have we learned
- Understand how to find angle and arc measures.
- Understand how to find an angle measure inside a circle.
- Understand how to find an angle measure outside a circle.
- Understand how to solve a real–word problem.
![Angle Triangle Similarity](/_next/image?url=https%3A%2F%2Fwww.turito.com%2Flearn-internal%2Fwp-content%2Fuploads%2F2022%2F09%2FAngle-Triangle-Similarity-1-1.png&w=1920&q=50)
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