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

• Central Angles
• 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°.

• 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 $\overline{AD}$

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.

• 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)

• 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°

• 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).

• Angles of Intersecting Chords Theorem
• Theorem and Formula
• 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: $\frac{1}{2}(144+90)=117$
• $\frac{1}{2}(234)=117$

• 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

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°

• 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°

• 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°.

• 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:  $62=\angle RST$

• 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: $\widehat{JK}=29$