1. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Many arches that you see in structures are semicircular, but Chinese builders long
ago discovered that arches don’t have to have this shape.
The Zhaozhou bridge, was completed in 605 A.D.
It is the world’s first stone arched bridge in the shape of a minor arc, predating other minor-arc arches by about 800 years.
In this lesson you’ll discover properties of arcs and the angles associated with them.
JRLeon Geometry Chapter HGHS

2. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Inscribed Angles
Given: Central Angle BOC and Inscribed Angle BAC and Chord AB  Chord AC
Show: BAC is half the measure of BOC
Chord AB  Chord AC (Given)
Construct Radius AO
We know that AO = CO = BO (Radii)
JRLeon Geometry Chapter HGHS

3. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Inscribed Angles
We see that we have two Isosceles triangles
The base angles of an Isosceles triangle are congruent
So BAC = x + y x
BOA = 180 – 2x
COA = 180 – 2y
180 – 2x O
BOC = 360° - BOA - COA
180 – 2y
BOC = 360° - (180°-2x) - (180°-2y) y
BAC = BOC / 2 x y
BOC = 360° - 180°+2x - 180°+2y
BOC = 360° - 360°+2x + 2y
BOC = 2x + 2y
BOC = 2(x + y)
If BOC = 2(x + y) and BAC = x + y
Then BOC = 2(BAC )
Showing that BOC / 2 = BAC
Or BAC = BOC / 2 = BC/2
JRLeon Geometry Chapter HGHS

4. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Inscribed Angles
BAC = BOC / 2
JRLeon Geometry Chapter HGHS

5. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Inscribed Angles Intercepting the Same Arc
JRLeon Geometry Chapter HGHS

6. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Angles Inscribed in a Semicircle
Next, lets look at the property of angles inscribed in semicircles. This will lead you to a third important conjecture about inscribed angles.
Arc AB has a measure of 180°
The measure of the inscribed angle is one-half the size of the intercepted arc.
Therefore, the measure of each of the inscribed angles is 90°.
JRLeon Geometry Chapter HGHS

7. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Cyclic Quadrilaterals
A quadrilateral inscribed in a circle is called a cyclic quadrilateral. Each of its angles is inscribed in the circle, and each of its sides is a chord of the circle. A D C B
Lets look at the two arcs DAC and CBD.
Arc DAC = twice angle B = 2B
Arc CBD = twice angle A = 2A
The measure of Arc CBD PLUS the measure of Arc DAC = 360°
2A + 2B = 360°
2(A + B) = 360°
(A + B) = 180°
JRLeon Geometry Chapter HGHS

8. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Arcs by Parallel Lines
Lets look at arcs formed by parallel lines that intersect a circle.
A line that intersects a circle in two points is called a secant. A secant contains a chord of the circle, and passes through the interior of a circle, while a tangent line does not. A secant is a line while a chord is a segment.
With segment AD, we have a transversal.
BAD = CDA (Alternate Interior Angles)
Therefore, the measure of arc AC = the measure of arc BD.
JRLeon Geometry Chapter HGHS

9. JRLeon Geometry Chapter 6..3 HGHS
6.3- Arcs and Angles
Lesson 6.3
Pages : Problems 1 thru 16
JRLeon Geometry Chapter HGHS