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6.5 – Circles, Angles, and Proofs

Key Terms

  • Inscribed Angle – an angle formed by two chords of a circle that share an endpoint.
  • Intercepted Arc – a part of a circle (an arc) that is cut off from the rest of the circle’s circumference by lines or segments intersecting the circle.

 


Review

  • Doubling
    • To make twice as many
    • To make twice the size of

 

  • Parts of a Circle
    • Central angles have sides called radii
    • Arcs have endpoints on the circle

GeoB 6.5 Parts of a Circle

 

  • Central Angles
    • Formed by 2 radii
    • Create a minor arc

 

  • Remote Exterior Angle Theorem of a Triangle
    • The remote exterior angle of a triangle equal the sum of the two interior remote angles
    • Ex. If the remote interior angles are 45° and 65°, the measure of the remote exterior angle would be their sum: 110°.

GeoB 6.5 Remote Ext Angle Review

 

  • Radii
    • A radius’ length is half the diameter’s length
    • All radii of the same circle are congruent
    • 2 radii can form an isosceles triangle with a corresponding chord
      • Ex. Radii \overline{BD} and \overline{AB} form an isosceles triangle with chord \overline{AD}

GeoB 6.5 Isosceles Triangle

 


Notes

  • Inscribed Angles
    • Formed by two chords that meet at one endpoint
    • The angle is inside the circle
    • The measure of an inscribed angle in a circle is half the measure of the intercepted arc.

GeoB 6.5 Inscribed Angle Img GeoB 6.5 Inscribed Central Angle

GeoB 6.5 Inscribed Angle Arc Ex

GeoB 6.5 Inscribed Angle Rule

 

  • Inscribed Angles vs Central Angles
    • Inscribed angles have half the measure of the intercepted arc
    • Central angles have the same measure as the intercepted arc
    • A central angle may have the same intercepted arc as an inscribed angle (see image below)

GeoB 6.5 Inscribed Central Angle Tool

 

  • Isosceles Triangles Within a Circle
    • Since 2 radii and a corresponding chord create an isosceles triangle, the 2 base angles are the same measure.
    • The inscribed angle and central angle are related through the triangle’s “remote exterior angle” theorem.
      • Ex. Radii \overline{BD} and \overline{AB} form an isosceles triangle with chord \overline{AD}
      • The base angles of the isosceles triangle measure 37° each.  They are remote interior angles from \angle ABC.
      • Since the sum of the two remote interior angles equals the measure of the remote exterior angle, 35° + 35° = 70°GeoB 6.5 Isosceles Triangle Angles

 

  • Proof: An intercepted arc is twice the measure of its inscribed angle

 

  • Semicircles are arcs that measure 180°, so their inscribed angles measure 90°
    • Semicircles have a corresponding chord called a diameter (which is equal to 2 radii).

GeoB 6.5 Semicircle Angles

 


  • Angles of Intersecting Chords Theorem
    • Theorem and FormulaGeoB 5.6 Angles of Intersecting Chords Theorem
    • Intersecting chords form a pair of congruent, vertical angles.
    • A chord is a segment, and a segment is part of a line.
    • So, intersecting chords create linear pairs and vertical angles.
    • Theorem: The measure of each vertical angle is half the SUM of the measures of the intercepted arcs.
      • Ex. The vertical angles in the example below are 117° each.  Their intercepted arcs are 144° and 90°.
        • Formula applied: \frac{1}{2}(144+90)=117
        • \frac{1}{2}(234)=117

GeoB 6.5 Intersecting Chords Rule

 

  • Formula for Angles of Intersecting Chords Theorem
    • \frac{1}{2}(Sum-of-Intercepted-Arcs)=Vertical-Angle-Measure
    • Watch the video and copy the proof for this formula in the template below

Geometric Proof Template

 


Examples

  • Ex 1. What is the measure of \widehat{JL}?
    • Since an inscribed angle is 1/2 the arc it intercepts, the arc is 168°

GeoB 6.5 Ex 01

 

  • Ex 2. What is the measure of \widehat{JKL}?
    • Circles measure 360°, so subtract the intercepted arc (140°) measure from 360° to find the answer of the major arc.
    • Answer: 360° – 140° = 220°

GeoB 6.5 Ex 02

 

  • Ex 3. What is the measure of \angle RST?
    • Since the inscribed angle is half the measure of the intercepted arc (which is 94°), the angle measures 47°.

GeoB 6.5 Ex 03

 

  • Ex 4. What is the measure of \angle RST?
    • Apply the formula: \frac{1}{2}(\widehat{PQ}+\widehat{RT})=\angle RST
    • \frac{1}{2}(\widehat{47}+\widehat{77})=\angle RST
    • \frac{1}{2}(124)=\angle RST
    • Answer:  62=\angle RST

GeoB 5.6 Q2 Ex01

 

  • Ex 5. What is the measure of arc \widehat{JK}?
    • Apply the formula: \frac{1}{2}(\widehat{MN}+\widehat{JK})=Vertical-Angle
    • Substitute what you know: \frac{1}{2}(\widehat{45}+\widehat{JK})=37
    • Multiply both sides by 2 to eliminate the fraction: 2(\frac{1}{2}(\widehat{45}+\widehat{JK})=37(2)
    • Simplify: \widehat{45}+\widehat{JK}=74
    • Subtract 45 on both sides: \widehat{JK}=29

GeoB 5.6 Q2 Ex02

 


 

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