## 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.

This proof will consist of two parts. The first and second equations of the theorem will be proved one at a time.

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

**Solution: **

Theorem10.11: One half the measure of the intercepted arc.

m∠1 = SMALL ARC / 2

= 1/2 (130^{0})

= 65

∴ m∠1 = 65^{0}

Line M is tangent to the circle. Find the measure of the red angle or arc.

**Solution:**

Theorem10.11: One half the measure of the intercepted arc

m KJL/2= 125^{0 …………..}

m KJL = 2.(125^{0})

= 250^{0}

∴ m KJL= 250^{0}

**Guided practice for Example 1:**

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 (210^{0}) …….

= 105^{0}

∴ m∠1 = 105^{0}

**2. mRST**

**Solution:**

Theorem10.11: One half the measure of the intercepted arc.

mRST / 2=98^{0…………}

m = 2(98^{0})

= 196^{0}

∴ mRST = 196^{0}

**3. mXY**

**Solution: **

mXY/2=80^{0}

m = 2(80^{0})

= 160^{0}

∴ mXY = 160^{0}

### Intersecting Lines and Circles

If two lines intersect a circle, there are three places where the lines can intersect.

- On the circle

- Inside the circle

- Outside the circle

- 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.

**Example 2:**

Find the value of *x*.

**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.

**Example 3:**

- Find the value of
*x*.

**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

**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.*

**Solution:**

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.)

### Let’s Check your Knowledge

Find the value of *x*. Any lines that appear to be tangent can be assumed to be tangent.

2

3.

4.

5.

6.

### Answers

## Exercise

- Line m is tangent to the circle. Find the indicated measure.

i. m∠1

- m∠RST

- Find the value of
*x.*

- Find the value of
*x.*

- Find the value of the variable.

- Find the value of x.

### Concept Map

### 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.

#### Related topics

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>#### How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>
Comments: