Geometry B, Assignment 7. Theorems (1)

Given: AB is the diameter of circle O. AC and BD are tangents Prove: AC || BD

1. AB is diameter of circle O AC and BD are tangents : Given 2. AB ⊥ AC, AB ⊥ BD : Radius ⊥ to tangent 3. AC || BD : 2 lines ⊥ to same line are ||

Given: AE is tangent to circle R at point A and to circle S at point E Prove: <R = <S

1. AE tangent to circle R at A and to circle S at E : Given 2. RA ⊥ AE , SE ⊥ AE : Radius ⊥ to tangent 3. RA || SE : 2 lines ⊥ to same line are || 4. <R = <S : If lines ||, alt. interior <'s are =

Given: Two concentric circles with AB tangent to smaller circle at R Prove: AR = RB

1. Draw OA, OB : Auxiliary lines 2. OR ⊥ AB : Radius ⊥ to tangent 3. OR = OR : Reflexive 4. OA = OB : Radii of same circles are = 5. rt (triangle) AOR = rt (triangle) BOR : HL 6. AR = RB : CPCTE

Given: AB, CD are diameters Prove: BD = CA

1. OD = OC , OB = OA : Radii of same circles are = 2. m <DOB = m <COA : Vertical <'s are = 3. triangle DOB = triangle COA : SAS 4. BD = CA : CPCTE

Given: Points R, S, T, Q on circle O Prove: m (arc) RS + m (arc) ST + m (arc) TQ = m (arc) RQ

1. Points R, S, T, Q on circle O : Given 2. m (arc) RS + m (arc) ST = m (arc) RT m (arc) RT + m (arc) TQ = m (arc) RQ : Arc addition 3. m (arc) RS + m (arc) ST + m (arc) TQ = m (arc) RQ : Substitution

Given: lines from point P are tangent to R at points A and B Prove: PA = PB

1. RA = RB : Radii of same circles are = 2. RP = RP : Reflexive 3. RA ⊥ AP , RB ⊥ PB : Radius ⊥ to tangent 4. rt (triangle) RAP = rt (triangle) RBP : HL 5. PA = PB : CPCTE

Given: RS tangent to circle A and circle B at points R and S Prove: AR || BS

1. RS tangent to circle A and circle B at points R and S : Given 2. AR ⊥ RS, BS ⊥ RS : Radius ⊥ to tangent 3. AR || BS : 2 lines ⊥ to same line are ||

Given: SW is diameter of R m (arc) ST = 30* Prove: m <RTW = 15*

1. m (arc) TS = 30 : Given 2. m <SRT = 30 : Central <'s = arcs 3. < T = <W : Base <'s of Isosceles triangle are = 4. m <SRT = <T + <W : Exterior < = Sum of remote interior <'s 5. 30* = 2 (m<t) : Substitution 6. (m<t) = 15* : Division

Given: m < XOY = m < WOV m (arc) YZ = m (arc) ZW Prove: m (arc) XZ = m (arc) ZV

1. m <XOY = m < WOV m YZ = m ZW : Given 2. m <YOZ = m <ZOW : Central <'s = arcs 3. m <XOY + m <YOZ = m <WOV + m <ZOW : Addition property of equality 4. m <XOZ = m <ZOV : Angle addition 5. m (arc) XZ = m (arc) ZV : m of arc = m of central <

Diameters AB and CD of circle K intersect such that m ∠ BKD = 100°. The measure of arc AC is:

100*

m (arc) PQR =

180*

m (arc) PSQ =

312*

m (arc) PQ =

48*

In circle O, central angle AOB has a measure of 90°. Which of the following is not true?

Arc AB is a semicircle

Arc RQ is a:

Minor arc

The theorems in this lesson relate to all of the following:

Tangents, arcs, and chords