Angle and Arc Measures

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

1.  

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.

Solution:

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

m KJL/2= 125^{0     …………..}

m KJL  = 2.(125⁰)                               

= 250⁰

∴ m KJL= 250⁰

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 (210⁰) …….                   

= 105⁰                  

∴ m∠1 = 105⁰

2.   mRST

Solution:

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

mRST / 2=98^{0…………}                

m  = 2(98⁰)                            

= 196⁰                  

∴ mRST  = 196⁰

3.  mXY

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

• 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:

1. 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.

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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.