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 6..3 HGHS
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 6..3 HGHS
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 6..3 HGHS
JRLeon Geometry Chapter 6..3 HGHS 6.3- Arcs and Angles Inscribed Angles BAC = BOC / 2 JRLeon Geometry Chapter 6..3 HGHS
JRLeon Geometry Chapter 6..3 HGHS 6.3- Arcs and Angles Inscribed Angles Intercepting the Same Arc JRLeon Geometry Chapter 6..3 HGHS
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 6..3 HGHS
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 6..3 HGHS
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 6..3 HGHS
JRLeon Geometry Chapter 6..3 HGHS 6.3- Arcs and Angles Lesson 6.3 Pages 327-328: Problems 1 thru 16 JRLeon Geometry Chapter 6..3 HGHS