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Interior Angles We learned about two, now here comes the third

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Central Angle & Inscribed Angle radius chord

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Intersecting Chords

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Definition of Intersecting Chords When two chords cross each other within a circle, their intersection creates four angles. Measuring the angles created by intersecting chords How can we measure these angles? The vertex is not in the center, nor is it on the circle. 1 2 3 4 A B D C

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First we see that ∡ 1 and ∡ 3 are opposite angles, so ∡ 1 = ∡ 3 ∡ 2 and ∡ 4 are also opposite angles, so ∡ 2 = ∡ 4 This means there are really only two angle measures to find! Measuring the angles created by intersecting chords 1 2 3 4 1 2 A B D C

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How to Find ∡ 1 1 2 1 2 A B D C ½ DC ½ AB Make a triangle with ∡ 2, ∡ B, and ∡ C. ∡ B is an inscribed angle capturing DC ∡ C is an inscribed angle capturing AB ∡ B = ½mDC and ∡ C = ½mAB

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How to Find ∡ 1 1 2 1 2 A B D C ½ DC ½ AB Then since it is a triangle, m ∡ B + m ∡ C + m ∡ 2 = 180 ̊ We notice that ∡ 1 and ∡ 2 are a linear pair, m ∡ 1 + m ∡ 2 = 180 ̊ This means… m ∡ 1+m ∡ 2=m ∡ B+m ∡ C+m ∡ 2 Cancel like terms to get m ∡ 1= m ∡ B + m ∡ C Which means that m ∡ 1= ½mAB + ½mDC.

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The formula for finding the measure of an angle created by intersecting chords m∡1 =½mAB+ ½mCD m∡1 =½(mAB+ mCD) Which can also be written as

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Let’s Try Some Examples

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A B D C 1 1 m∠1=½( mAB+ mCD )

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A B D C m∠1=½( mAB+ mCD ) 30° 40° + 30° )= 35° 1 1

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A B D C 2 2 m∠2=½( mAD+ mBC ) 40° 35° m∠1=35° 30°

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A B D C 2 2 m∠2=½( mAD+ mBC ) 210° 80° 210°+ 80° )= 145° 40° 35° 30°

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A B D C 4 m∠5=½( mAD+ mBC ) 170° 90° 170°+ 90° )= 130° 5 m∠4 = 180° – 130° = 50°

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In Class Assignment Watch the screen, answer the questions in your notebook (10 questions total)

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Section 6.4 Inscribed Polygons

Section 6.4 Inscribed Polygons

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