Lesson 10.2 Arcs and Chords

Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Minor Arc formed from a central angle less than 180° minor arc

Major Arc formed from a central angle that measures between 180 ° ° major arc

Semicircle formed from an arc of 180 ° Half circle! Endpoints of an arc are endpoints of the diameter

Naming Arcs How do we name minor arcs, major arcs, and semicircles??

Minor Arc Named by the endpoints of the arc. Minor Arc: AB or BA

Major Arc Named by the endpoints of the arc and one point in between the arc Major Arc: ACB or BCACould we name this major arc BAC?

Semicircle Named by the endpoints of the diameter and one point in between the arc mABC = 180°

Example

Measuring Arcs A Circle measures 360 °

Measure of a Minor Arc Measure of its central angle 95° m AB=95 °

Measure of a Major Arc difference between 360° and measure of minor arc 95° mACB=360°– 95° = 265°

Arc Addition Postulate Measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. What is the measure of BD? m BD=100 °

Example for #1-10

Congruent Arcs Two arcs of the same circle or congruent circles are congruent arcs if they have the same measure. AB is congruent to DC since their arc measures are the same.

Theorem 10.4 Two minor arcs are congruent iff their corresponding chords are congruent. Chords are congruent

Example 1 Solve for x 2xX+40

Theorem 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If DE = EF, then DG = GF

Example. Find DC. m DC = 40º

Theorem 10.6 If one chord is a perpendicular bisector to another chord, then the first chord is a diameter. Since AB is perpendicular to CD, CD is the diameter.

Example. Solve for x. x = 7

Theorem 10.7 Two chords are congruent iff they are equidistant from the center. Congruent Chords

Example. Solve for x. x = 15