Answer key. when inscribed angles intercept equal arcs, they are congruent an angle inscribed in a semi-circle is a. All right angles are congruent
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1 Name Answer key Lesson 5.5 Circle Proofs Date CC Geometry Do Now Day 1 : Circle O, m ABC 42 o Find: m ADC Do Now Day 2 Name two valid statements and reasons: : Circle O, arc BA congruent to arc DE <ACB=<DFE <ABC and <DEF are right angles <ABC=<DEF when inscribed angles intercept equal arcs, they are congruent an angle inscribed in a semi-circle is a right angle All right angles are congruent
2 1. In a circle all radii are congruent 2. If a radius (or part of a radius) is perpendicular to a chord, then it bisects the chord. If AD BC, then CE BE 3. If a radius (or part of a radius) bisects a chord, then it is perpendicular to the chord. If CE BE, then AD BC 4. Congruent central angles have congruent chords. If BEA CED, then BA CD. 5. Congruent arcs have congruent central angles. If arcba arccd, then BEA CED 6. Congruent chords have congruent arcs. If BA CD, then arcba arccd. equidistant from the center of a circle, then they are congruent. If OB OE, then AC DF. 10. If two chords are congruent, then they are equidistant from the center. If AC DF, then OB OE. 11. If two segments from the same exterior point are tangent to a circle, then they are congruent. If AB and CB are tangent to circle D, then AB CB. 12. If two inscribed or tangent-chord angles intercept the same arc, then they are congruent. 13. If arcs are congruent, then the inscribed angles that intercept those arcs are congruent. 7. An angle inscribed in a semicircle is a right angle. 8. In a circle, if two arcs are congruent, then the chords that intercept them are parallel. If AB CD then, arcac = arc BD 14. If a tangent is drawn to a radius or a diameter, then the angle formed is a right angle. 9. If two chords are Practice Examples
3 1. circle A with AD BC, prove ACE ABE circle A with AD l BC given AE=AE reflexive property <CEA and <BEA are right angles perpendicular lines form right angles <CEA=<BEA all right angles congruent AC=AB in a circle, all radii are congruent ACE and ABE are right triangles a triangle with one right angle is a right triangle ACE = ABE HL=HL 2. : BO EO AOC DOF BO=EO given AO=CO=DO=FO in a circle, all radii are congruent AC=DF if 2 chords are equidistant from the center of a circle then they are congruent AOC= DOF SSS=SSS
4 3. In circle O, PC and PB are tangents, COP BOP In circle O, PC and PB are tangents given CP=BP when two tangents are drawn to the same external point then they are congruent OC=OB in a circle, all radii are congruent OP=OP reflexive property COP= BOP SSS=SSS COP= BOP corresponding parts of congruent triangles are congruent 4. In the accompanying diagram, BR YD and BOD is the diameter of circle O. RBD YDB BR=YD and BOD is the diameter or circle O given <R and <Y are right angles In a circle, an inscribed angle in a semi-circle is a right angle <R=<Y all right angles are congruent BD=BD Reflexive Property RDB and YDB are right triangles a triangle with a right Angle is a right triangle RDB= YDB HL=HL
5 5. : chords AB, CD, AD, CB ADE ~ CBE Chords AB, CD, AD, CB <A=<C, <B=<D ADE CBE when two inscribed angles intercept the same arc then they are congruent AA=AA 6. : arc AB is congruent to arc CD BEC ~ FEG AB = CD <E=<E BCllAD if 2 arcs are congruent then the chords that intercept those arcs are parallel <CBE=<GFE when parallel lines are cut by a transversal corresponding angles are congruent BEC FEG AA=AA
6 7. : AC is tangent to circle O at A and circle P at C. OAB ~ PCB AC is tangent to circle O at A & circle P at C <OBA=<PBC <OAB and <PCB are right angles <OAB=<PCB vertical angles are congruent When a tangent line intercepts a radius a right angle is formed All right angles are congruent OAB PCB AA=AA 8. In the accompanying diagram of circle O, diameter AOB is drawn, tangent CB is drawn to the circle at B, E is a point on the circle, and BE ADC BEllADC s <AEB is a right angle s an angle inscribed in a semi-circle is a right angle ABE ~ CAB <CBA is a right angle when a tangent intercepts a radius a right angle is formed <CBA=<AEB All right angles are congruent <CAB=<ABE when parallel lines are cut by a transversal, alternate interior angles are congruent ABE CBA AA=AA
7 Homework Day 1 1. : Circle Q, PR ST PS PT Circle Q, PS l ST PR=PR <SRP and <TRP are right angles Perpendicular lines form right angles <SRP=<TRP All right angles are congruent PR bisects ST When part of a radius is perpendicular to a chord it bisects the chord SR=TR A bisector divides a segment into 2 congruent parts PRS= PRT SAS=SAS PS=PT corresponding parts of congruent triangles are congruent 2. In circle O, DE FE D F In circle O, DE=FE DE=FE Congruent arcs have congruent chords OE=OE OD=OF In a circle, all radii are congruent DOE= FOE SSS=SSS <D=<F Corresponding parts of congruent triangles are congruent
8 3. : Circle O, OC AE, AB DE OB OD Circle O, OC l AE, AB=DE <OFB and <OFD are right angles perpendicular lines form right angles <OFB=<OFD OF=OF OC bisects AE All right angles are congruent when a radius is perpendicular to a chord it bisects the chord BF=FD Subtraction postulate OFB= OFD SAS=SAS OB=OD corresponding parts of congruent triangles are congruent 4. : AB and CD intersect at O, the center AC BD AB and CD intersect at O <COA=<DOB OC=OD=OA=OB Vertical angles are congruent in a circle, all radii are congruent COA= BOD SAS=SAS AC=BD Corresponding parts of congruent triangles are congruent
9 Homework Day 2 1. s AC=AD, B is the Midpoint of CD AC=AD AB=AB CB=DB ABC= ABD s congruent arcs have congruent chords A midpoint divides a segment into two congruent parts SSS=SSS 2. s s <ACB=<DBC BC=BC <A=<D when two inscribed angles intercept the same arc, they are congruent ACB= DBC AAS=AAS
10 3. s s Tangent AD, diameter CD Secant AC <C=<C <ADC is a right angle when a tangent meets a diameter, a right angle is formed <DBC is a right angle an angle inscribed in a semi-circle is a right angle <ADC=<DBC all right angles are congruent ADC DBC AA=AA 4. s s ABllDC DC = DC AD=BC parallel lines intercept congruent arcs <BDC=<ACD when inscribed angles intercept equal arcs, they are congruent <DAC=<CBD when inscribed angles intercept the same arc, they are congruent ACD= BDC AAS=AAS
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