P r o o f s Given: TA ≊ TO; PAD ≊ POD Prove: PT is the ⊥ Bisector of AO Boat Shape Proof [Boat Shape Prf] Statement Reason 1. TA ≊ TO 1. Given 2. TA = TO 2. 3. T is on the ⊥ Bisector of AO 3. PtsOn⊥Bis IFF Eqd 4. 4. Given PAD ≊ POD s ≊ IFF Meas = 5. PA ≊ PO 5. Isos△SidesIFF⊀s 6. PA = PO 6. 7. P is on the ⊥ Bisector of AO 7. PtsOn⊥Bis IFF Eqd 8. PT is the ⊥ Bisector of AO 8. 1LineThruPt ⊥ s ≊ IFF Meas = Given: XZY ≊ XYZ; X is the midpoint of VW; △XVZ and △XWY are right △s Prove: XVC ≊ XWY Embedded Shape Proof Statement 1. XZY ≊ [Embedded Shape Prf] Reason XYZ 1. Given 2. XZ ≊ XY 2. Isos △SidesIFF s 3. X is the midpoint of VW 3. Given 4. VX ≊ WX 4. Segment Midpoint 5. △ XVZ and △XWY are right △s 6. △ XVZ ≊ △XWY ⁂ 5. Given 6. HL Right △ ≊ Given: P is the circumcenter of △XYZ Prove: P is the same distance from X, Y and Z Circumcenter Theorem Proof [CircumctrThm Prf] Statement 1. P is the circumcenter of △XYZ Reason 1. Given 2. P lies on the perpendicular bisectors of XY, YZ and XZ 2. △ Circumcenter 3. PX = PY 3. Pt On⊥BisIFFEqd 4. PY = PZ 4. Pt On⊥BisIFFEqd 5. PX = PY = PZ 5. Transitive Prop 6. P is the same distance from X, Y and Z 6. Dist PointTo Line Given: m 2 = 90 ° Prove: 1 and 3 are complementary Acute Statement s RIght △ Proof [Acute s RIght△ Prf] Reason 1. m 2 = 90° 1. Given 2. m⊀1 + m 2 + m 3 = 180° 2. Sum Interior 3. m⊀1 + 90° + m 3 = 180° 3. Subst Prop 4. m⊀1 + 90° + m 3 - 90°= 180° - 90° 4. Subtr Prop 5. m 1 + m 3 = 90° 5. Combine Terms 6. 6. Complementary 1 and 3 are complementary s△ s Given: B is the midpoint of AE and CD Prove: △ABC ≊ △EBD Common Midpoint Proof [Common Midpt Prf] Statement Reason 1. B is the midpoint of AE 1. Given 2. B is the midpoint of CD 2. Given 3. AB ≊ EB 3. Seg Midpoint 4. CB ≊ DB 4. Seg Midpoint 5. ABC ≊ EBD 5. Vert ical s ≊ 6. ABC ≊ EBD 6. SAS ≊ Given: 1 ≊ 3; A ≊ B; AF ≊ FB Prove: △AFE ≊ △BFD Half Star Proof [Half Star Prf] Statement Reason 1. 1 ≊ 3 1. Given 2. 1= 3 2. 3. 2= 2 4. 1+ 2= 5. AFE = 1+ 6. BFD = 3+ 7. AFE = BFD 7. Substitution Prop 8. AFE ≊ BFD 8. 9. A≊ 3. Reflexive Prop 3+ 2 2 B 10. AF ≊ FB s ≊IFF=Measure 11. △AFE ≊ △BFD 2 4. Addition 5. Addition 6. Addition s ≊IFF=Measure 9. Given 10. Given 11. ASA △ ≊ Given: AG ≊ EG; GM is a median of △AGE Prove: △MAG ≊ △MEG Statement Median Proof [Median Prf] Reason 1. AG ≊ EG 1. Given 2. GM ≊ GM 2. Reflexive Prop ≊ 3. GM is a median of △AGE 3. Given 4. M is the midpoint of AE 4. Def Median 5. AM ≊ EM 5. Def Midpoint 6. △MAG ≊ △MEG 6. SSS △ ≊ Given: Exterior s 1 and Prove: Exterior Angle Theorem Proof Statement 4; Interior 3 1+ 3= 4 [Exterior Thm Prf] Reason 1. Exterior 4 1. Given 2. Interior 1 and 3. 4 and 2 are a linear pair 3. Linear Pair 4. 4 and 2 are supplementary 4. Linear Pair Supp 5. 4 + 2 = 180° 5. Supplementary 6. 1+ 2+ 3 = 180° 6. Sum 7. 1+ 2+ 3= 4 + 2 7. Transitive Prop 8. 1+ 2+ 3- 2= 8. Subtraction Prop 9. 1+ 3= 4 3 ⁂ 2. Given 4 + 2- 2 s△ 9. Combine Terms s Given: 1 ≊ 2; 3 ≊ 4 Prove: △RAY ≊ △BAN Semaphore Proof [Semaphore Prf] Statement Reason 1. 1. Given 1≊ 2 2. RA ≊ BA 2. Isos△ SidesIFF s 3. 3. Given 3≊ 4 4. Isos△ SidesIFF s 4. YA ≊ NA 5. RAY ≊ BAN 6. △RAY ≊ △BAN 5. Vertical 6. SAS △ ≊ s ≊ Given: △ABC is isosceles; DE || AB Prove: △DEC is isosceles Rock Climbing Proof Statement 1. △ABC is isosceles [RockClimbingPrf] Reason 1. Given 2. AC ≊ BC 2. Isosceles △ 3. m A = m B 3. Isos△ SidesIFF s 4. DE || AB 4. Given 5. A ≊ 1 5. Corresponding s 6. B≊ 2 6. Corresponding s 7. 1≊ 2 8. DC ≊ EC 9. △DEC is isosceles 7. Transitive Prop ≊ 8. Isos△ SidesIFF s 9. Isosceles △ Given: △TJC an isosceles △; △TKC an isoceles △ Prove: △TAC is an isosceles △ Nested Triangles Proof [Nested △s Prf] Statement Reason 1. △TJC an isosceles △ 1. Given 2. TJ ≊ CJ 2. Isosceles △ 3. TJ = CJ 3. Segs ≊ IFF Lens= 4. J is the same distance from T and C 4. Distance 5. J is on the ⊥ bisector of TC 5. PtsEqdIFFOn⊥bis 6. △TKC an isosceles △ 6. Given 7. TK ≊ CK 7. Segs ≊ IFF Lens= 8. TK = CK 8. Isosceles △ 9. K is the same distance from T and C 9. Distance 10. K is on the ⊥ bisector of TC 10.PtsOn⊀ BisIFFEqd 11. Line segment AJKM is the ⊥ bisector of TC 11. 1LineThruPt⊥ 12. TA = CA 12. PtsEqdIFFOn⊥bis 13. TA ≊ CA 13. Segs ≊ IFF Lens= 14. TAC is an isosceles △ 14. Isosceles △ Given: 1≊ 2; G≊ P; GI ≊ PW Prove: BG || PW Horizon Proof Statement Reason 1. 1≊ 2. 1 and 3 are a linear pair 2. Linear Pair 3. 1 and 3 are supplementary 3. Linear Pair Supp 4. 2 and 4 are a linear pair 4. Linear Pair 5. 2 and 4 are supplementary 5. Linear Pair Supp 6. ⊀3 ≊ 7. G≊ 2 [Horizon Prf] 1. Given 4 6. SuppsTo ≊ s ≊ P 7. Given 8. GI ≊ PW 9. △BIG ≊ △WUP 10. 5≊ 11. BG || PW 6 ⁂ 8. Given 9. ASA △ ≊ 10. Corr Pts ≊ △ ≊ 11. AIA ≊ Lines|| Given: △RST is equiangular Prove: △RST is equilateral Equiangular Equilateral Proof [Eq EqLat Prf] Statement Reason 1. △RST is equiangular 1. Given 2. 2. Equiangular S≊ T 3. RS ≊ RT 3. Isos△SidesIFF s 4. 4. Equiangular R≊ S 5. ST ≊ RT 5. Isos△SidesIFF s 6. ST ≊ RS ≊ RT 6. Transitive Prop≊ 7. △RST is equilateral ⁂ 7. Equilateral Given: RS ≊ CS; ⊀ 2 ⊀3 Prove: RY ≊ CY Statement Barred Door Proof [Barred Door Prf] Reason 1. RS ≊ CS 1. Given 3. 3. Isos△ SidesIFF 2. △RSC is an isosceles △ 1≊ 4 2. Isosceles △ 4. YS ≊ YS 4. Reflexive Prop ≊ 5. 2≊ 3 5. Given 6. 5≊ 6 6. Third 7. △RYS ≊ △CYS 8. RY ≊ CY Thm 7. ASA △ ≊ 8. Corr Pts ≊△ ≊ Given: R is the incenter of △JKL Prove: R is the same distance from the sides of △JKL Incenter Theorem Proof Statement [Incenter Thm Prf] Reason 1. Draw RX ⊥ JK 1. 1LineThruPt ⊥ 2. RX is the distance between R and JK 2. Dist Pont To Line 3. Draw RY ⊥ JL 3. 1LineThruPt ⊥ 4. RY is the distance between R and JL 4. Dist Pont To Line 5. Draw RZ ⊥ KL 5. 1LineThruPt ⊥ 6. RZ is the distance between R and KL 6. Dist Point To Line 7. R is the incenter of △JKL 8. R lies on the 9. ⊀YLR bisectors of J, K and 7. △ Incenter L ⊀ZLR 8. Dist Point To Line 9. ⊀ Bisector 10. LR LR 10. Reflexive Prop 12. RY RZ 12. CorrPts △ 11. △YLR △ZLR 13. ⊀XKR ⊀ZKR 14. RK RK 16. RX RZ 17. RX RY 15. △XKR △ZKR 11. HA Right △ 13. ⊀ Bisector 14. Reflexive Prop 15. HA Right △ 16. CorrPts △ RZ 18. R is the same distance from the sides of △JKL 17. Transitive Prop 18. Dist Pont To Line Given: RS ⟂ RO; AE ⟂ AN; OE ≊ NS; RS ≊ AE Prove: △ROS ≊ △ANE Hour Glass Proof [Hour Glass Prf] Statement Reason 1. RS ⟂ RO 1. Given 2. AE ⟂ AN 2. Given 3. ORS is a right 3. ⟂ Lines 4. NAE is a right 4. ⟂ Lines 5. OE ≊ NS 5. Given 6. OE = NS 6. Segs ≊ IFF Lens= 7. ES = ES 7. Reflexive Prop 8. OE + ES = NS + ES 8. Seg Addition 9. OE + ES = OS 9. Seg Addition 10. NS + ES = NE 10. Seg Addition 11. NE = OS 11. Substitution Prop 12. NE ≊ OS 12. Segs ≊ IFF Lens= 13. RS ≊ AE 13. Given 14. △ROS ≊ △ANE 14. HL Rt ≊ Given: DE = RO; △OXE is Isosceles Prove: △DOX ≊ △REX Pyramid Proof [Pyramid Prf] Statement Reason 1. DE = RO 1. Given 2. DO = DE - OE 2. Seg Subtraction 3. RE = RO - OE 3. Seg Subtraction 4. RE = DE - OE 4. Substitution Prop 5. DO = RE 5. Transitive Prop 6. DO ≊ RE 6. Segs ≊IFF=Len 7. △OXE is isosceles 8. OX ≊ EX 9. 10. 11. XOE ≊ XEO DOX and DOX and REX and 13. REX and 10. DOX ≊ XOE are a linear pair XOE are supplementary XEO are a linear pair XEO are supplementary REX 11. △ DXO ≊ △REX 7. Given 8. Isosceles △ 9. Isos△ SidesIFF 10. Linear Pair 11. Supplementary 12. Linear Pair 13. Supplementary Supps To ≊ s ≊ SAS △ ≊ Given: IN ≊ IO; OWN ≊ NKO Prove: △WOZ ≊ △KNZ Folded Arms Proof [Folded Arms Prf] Statement Reason 1. IN ≊ IO 2. △ION is isosceles 3. ION ≊ INO 4. ON ≊ ON 5. OWN ≊ NKO 3. Isos△SidesIFF 5. Given 6. AAS △ ≊ 7. Corr Pts ≊△s 7 OW ≊ NK WZO ≊ 2. Isosceles △ 4. Reflexive Prop ≊ 6. △WON ≊ △KNO 8. 1. Given KZN 9. △WOZ ≊ △KNZ 8. Vertical 9. AAS △ ≊ s≊ s Given: BD Bisects EBF Prove: DE ≊ DF Two Heights Proof Statement Reason 1. BD bisects 2. EBD ≊ [Two Heights Prf] EBF FBD 1. Given 2. Bisector 3. Let DE be the altitude from D to EB 3. Dist Point To Line 4. DE ⟂ EB 4. LineThruPt ⟂ 5. 5. Perp Lines DEB = 90° 6. △DEB is a right △ 6. Right △ 7. Let DF be the altitude from D to BF 7. Dist Point To Line 8. DF ⟂ BF 8. LineThruPt ⟂ 9. 9. Perp Lines DFB = 90° 10. △DFB is a right △ 10. Right △ 11. BD ≊ BD 11. Reflexive Prop ≊ 13. DE ≊ DF 13. Corr Pts ≊ △s 12. △EBD ≊ △DFB 12. HA Right △ ≊ Given: AED ≊ ACD Prove: j ≊ k Obtuse Height Proof [Obtuse Height Prf] Statement Reason 1. Construct altitude EQ ⊥ AC 1. LineThruPoint ⊥ 2. ⊀EQA is a right ⊀ 2. ⊥ Lines 3. Construct altitude JR ⊥ EA 3. LineThruPoint ⊥ 4. ⊀CRA is a right ⊀ 4. ⊥ Lines 5. ⊀AED ≊ ⊀ACD 5. Given 6. △EAC is isosceles 6. Isos△ SidesIFF s 7. EA ≊ AC 7. Isosceles △ 8. ⊀1 ≊ ⊀2 8. Vertical ⊀s ≊ 10. ⊀C = ⊀E 10. Corr Pts ≊ △s 12. j ≊ k 12. Corr Pts ≊ △s 9. △EQA ≊ △CRA 11. △CRE ≊ △EQC 9. HA Right △ ≊ 11. HA Right △ ≊ Given: AB ⊥ BC; AE ⊥ DE; ≊ 2; AB ≊ AE Prove: AC ≊ AD Kite Flight Proof [Kite Flight Prf] Statement Reason 1. AB ⟂ BC 1. Given 2. 2. ⟂ Lines 3 is a right 3. AE ⟂ DE 3. Given 4. 4 is a right 4. ⟂ Lines 5. 3= 4 5. Right 6. 3 ≊ 4 6. 7. 1 ≊ 2 7. Given 8. AB ≊ AE 8. Given 10. AC ≊ AD 10. Corr Pts ≊ △s 9. △ACB ≊ △ADE s≊ s ≊ IFFMeasures= 9. ASA △ ≊ 1 Given: △QRS is isosceles; RPT ≊ RTP Prove: PQ ≊ TS Triangle Hat Proof Statement 1. RPT ≊ [△ Hat Prf] Reason RTP 1. Given 2. PR ≊ TR 2. Isos△ SidesIFF s 3. PR = TR 3. Segs ≊ IFF Lens= 4. △QRS is isosceles 4. Given 5. QR ≊ SR 5. Isosceles △ 6. QR = SR 6. Segs ≊ IFF Lens= 7. PQ + QR = PR 7. Segment Addition 8. TS + SR = TR 8. Segment Addition 9. PQ + QR = TS + SR 9. Transitive Prop 10. PQ + SR = TS + SR 10. Substitution Prop 11. PQ + SR - SR = TS + SR - SR 11. Subtraction Prop 12. PQ = TS 12. Combine Terms 13. PQ ≊ TS 13. Segs ≊ IFF Lens= Given: BL bisects QBP; BL || JP Prove: △PBJ is Isosceles Jungle Gym Proof Statement Reason 1. BL bisects QBP 2. BL || JP 1. Given 2. Given 3. QBL ≊ 4. J ≊ [Jungle Gym Prf] PBL QBL 5. J≊ 6. PBL ≊ 7. J≊ 8. BP ≊ BJ PBL BPJ BPJ 9. △PBJ is Isosceles 3. Bisector 4. Corr s 5. Transitive Prop≊ 6. Alt Int s 7. Transitive Prop≊ 8.Isos△ Sides IFF ⊀s 9. Isosceles △ Given: △MNO is isosceles; QM bisects NMO Prove: QM ⟂ NO Statement Paper Folding Proof [Paper Folding Prf] 1. △ MNO is isosceles 2. MN ≊ MO 3. QM bisects 4. MNQ ≊ NMO OMQ 5. MQ ≊ MQ 6. △NMQ ≊ △OMQ 7. NQM ≊ OQM Reason 1. Given 2. Isosceles △s 3. Given 4. Bisector 5. Reflexive Prop≊ 6. SAS △ ≊ 7. Corr Pts ≊ △s 8. m NQM = m OQM 8. 9. 9. Linear Pair NQM and OQM are a linear pair s ≊ IFFMeasures= 10. m NQM + m OQM = 180° 10. Linear Pair Supp 11. m NQM + m NQM = 180° 11. Substitution Prop 12. 2m NQM = 180° 12. Combine Terms 13. 2m NQM/2 = 180°/2 13. Division Prop 14. m NQM = 90° 14. Quotient 15. 15. Right NQM is a right 16. QM ⟂ MO ⁂ 16. ⟂ Lines Given: △MNO is isosceles; QM bisects NMO Prove: QM ⟂ NO Paper Folding Proof 2 Statement [Paper Folding Prf 2] Reason 1. △ MNO is isosceles 1. Given 2. QM bisects 2. Given NMO 3. MQ is ⟂ Bisector of NO 4. QM ⟂ MO 5. ⁂ 3. Isos △ 4. ⟂ Lines Bisector Given: △MLP is isosceles; N is the midpoint of MP Prove: LN ⊥ MP Statement Incline Proof [Incline Prf] Reason 1. △MLP is isosceles 1. Given 2. N is the midpoint of MP 2. Given 3. LN is the median from vertex L to midpoint N 3. △ Median 4. LN is the ⊥ Bisector of MP 4. Isos△ Median 5. LN ⊥ MP 5. Perp Bisector Given: AB ≊ CD; BFA ≊ BFE; DEC ≊ DEF Prove: FB ≊ ED Train Station Proof [Train Station Prf] Statement Reason 1. AB ≊ CD 1. Given 2. AC ≊ CA 2. Reflexive Prop ≊ 3. BFA ≊ BFE 3. Given 4. BFA and BFE are supplementary 4. Supplementary 5. BFA and BFE are right 5. ≊ Supp 7. DEC ≊ 8. DEC and DEF are supplementary 8. Supplementary 9. DEC and 9. ≊ Supp 6. △ABC ≊ △CDA 10. BFA ≊ DEF are right DEC s Rt 6. HL RIght △ ≊ DEF 11. △BFA ≊ △DEC 12. FB = ED s s s 7. Given s s Rt 10. Supplementary 11. AAS △ ≊ 12. Corr Pts ≊ △ s s s Given: C is equidistant from A and B; E is the midpoint of AB Prove: CE ⟂ AB Bicycle Rack Proof [BicycleRack Prf] Statement Reason 1. C is equidstant from A and B 1. Given 2. E is the midpoint of AB 2. Given 3. CA ≊ CB 3. Segment Midpoint 4. CE ≊ CE 4. Reflexive Prop≊ 5. AE ≊ BE 5. Segment Midpoint 6. △CAE ≊ △CBE CEB 6. SSS △ ≊ 7. Corr Pts ≊ △s 7. CEA ≊ 8. CEA and CEB form a linear pair 8. Adjacent 9. CEA and CEB are supplementary 9. Linear Pair Supp s 10. m CEA + m CEB = 180° 10. Supplementary 11. m CEA + m CEA = 180° 11. Substitution Prop 12. 2m CEA = 180° 12. Combine Terms 13. 2m CEA/2 = 180°/2 13. Division Prop 14. m CEA = 90° 14. Quotient 15. 15. Right CEA is a right 16. CE ⟂ AB 16. ⟂ Lines s Given: AB = AC; EC = ED; m B = 55 °; m D = 35 ° Prove: AC ⟂ CE Statement Tall Short Proof [Tall Short Prf] Reason 1. AB = AC 1. Given 2. EC = ED 2. Given 3. m B = 55° 3. Given 4. m D = 35° 4. Given 5. ACB is isosceles 6. EDC is isosceles 5. Isosceles △ 6. Isosceles △ 7. m ACB = m B 7. sides ≊ IFF s ≊ 8. m ECD = m D 8. Base 9. m ACB = 55° 9. Subst Prop 10. m ECD = 35° 10. Subst Prop 11. m ACB + m ACE + m ECD = 180° 11. Partition Postulate 12. 55° + m ACE + 35° = 180° 12. Subst Prop 13. 90° + m ACE = 180° 13. Combine Terms 14. 90° + m ACE - 90° = 180° - 90° 14. Subtr Prop 15. m ACE = 90° 15. Combine Terms 16. AC ⟂ CE 16. ⟂ Lines s≊ Given: △XZY Isosceles; XP the ⊀ bisector of the vertex angle Prove: XP the ⟂ bisector of the base Isosceles △ Bisector ⟂ Bisector Proof [Iso△ Bis⟂BisPrf] Statement Reason 1. △XZY is isosceles 2. XP is the Bisector of the vertex angle, 3. YXP PXZ ≊ 4. XP ≊ XP 5. △PXZ ≊ △YXP 6. ZPX ≊ YPX 1. Given ZXY 2. Given 3. Bisector 4. Reflexive Prop ≊ 5. SAS △ ≊ 6. Corr Pts ≊ △s 7. m ZPX = m YPX 7. 8. 8. Linear Pair ZPX and YPX are a linear pair s ≊ IFF=measure 9. m ZPX + m YPX = 180° 9. Linear Pair Supp 10. m ZPX + m ZPX = 180° 10. Substitution Prop 11. 2(m ZPX) = 180° 11. Combine Terms 12. 2(m ZPX)/2 = 180°/2 12. Division Prop 13. m ZPX = 90° 13. Quotient 14. XP ⟂ ZY 14. ⟂ Lines 15. ZP ≊ YP 15. Corr Pts ≊ △s 16. ZP = YP 16. 17. P is the midpoint of ZY 17. Seg Midpoint 18. XP is the ⟂ bisector of ZY 18. ⟂ Bisector s ≊ IFF=measure Given: AB ≊ AC; BX bisects B; CX bisects C Prove: BX ≊ CX Concave Proof [Concave Prf] Statement Reason 1. AB ≊ AC 1. Given 2. B ≊ 2. Isos△ SidesIFF s C 3. m B = m C 3. s ≊ IFFMeasures= 4. BX bisects B 4. Given 5. CX bisects C 5. Given 6. 1 ≊ 2 6. Bisector 7. 3 ≊ 4 7. Bisector 8. m 1 = m 2 8. s ≊ IFFMeasures= 9. m 3 = m 4 9. s ≊ IFFMeasures= 10. m A = m 1 + m 2 10. 11. m B = m 3 + m 4 11. 12. m A = m 2 + m 2 12. Substitution Prop 13. m B = m 4 + m 4 13. Substitutuon Prop 14. m A = 2(m 2) 14. Combine Terms 15. m B = 2(m 4) 15. Combine Terms 16. 2(m 2) = 2(m 4) 16. Transitive Prop 17. 2(m 2)/2 = 2(m 4)/2 17. Division Prop 18. m 2 = m 4 18. Quotient 19. 19. 2 ≊ 4 20. BX ≊ CX Addition Addition s ≊ IFFMeasures= 20. Isos△ SidesIFF s Given: IL the ⟂ bisector of HK Prove: Arrow Head 1 Proof Statement 2≊ 4 [ArrowHead 1 Prf] Reason 1. IL ⟂ the Bisector of HK 1. Given 2. HI = KI 2. ⟂ Bisector 3. KJ = HJ 3. PtOn⊥Bis=Dist 4. IJ = IJ 4. Reflexive Prop 5. △HIJ ≊ △KIJ 6. 1≊ 3 7. JL = JL 9. △HIL ≊ △KIL H≊ 6. Corr Pts ≊ △s 7. Reflexive Prop 8. HL = KL 10. 5. SSS △ ≊ K 8. PtOn ⟂ Bis 9. SSS △ ≊ 10.Corr Pts ≊ △s 11. m K = m 1 = m 2 11. Addition 12. m H = m 3 + m 4 12. Addition 13. m 3 + m 2 = m 3 + m 4 13. Substitution Prop 14. m 3 + m 2 - m 3 = m 3 + m 4 - m 3 14. Subtraction Prop 15. m 2 = m 4 15. Combine Terms 16. 16. 2 ≊ 4 s ≊IFF=Measure Given: IL ⟂ bisector of HK Prove: Statement Arrow Head 2 Proof 2≊ 4 [ArrowHead 2 Prf] Reason 1. IL ⟂ Bisector of HK 1. Given 2. HL ≊ KL 2. Pt on ⟂Bis IFF EqD 4. IL is the altitude of △HKL 4. △ Altitude 3. △HKL is isosceles 3. Isosceles △ 3 5. Isos △ Altitude 6. JL ≊ JL 6. Reflexive Prop ≊ 8. 8. CorrPts ≊△s 5. 1≊ 7. △ JHL ≊ △ JKL 2≊ 4 7. SAS △≊ Given: XZ = XY; ZXP ≊ YXP Prove: XP is the ⟂ bisector of ZY Angle Bisector Perpendicular Bisector Proof [ Bis ⟂ Bis Prf] Statement Reason 1. XZ = XY 1. Given 2. ZXP ≊ YXP 2. Given 3. XP ≊ XP 3. Reflexive Prop ≊ 5. ZP ≊ YP 5. Corr Pts ≊ △s 6. P is the midpoint of ZY 6. Segment Midpoint 4. △XZP ≊ △XYP 7. ZPX ≊ YPX 4. SAS △ ≊ 7. Corr Pts ≊ △s 8. m ZPX + m YPX = 180° 8. Addition 9. m ZPX + m ZPX = 180° 9. Substitution Prop 10. 2(m ZPX) = 180° 10. Combine Terms 11. 2(m ZPX)/2 = 180°/2 11. Division Prop 12. m ZPX = 90° 12. Quotient 13. XP ⟂ ZY 13. ⟂ Lines 14. XP is the ⟂ bisector of ZY 14. ⟂ Bisector Given: △ACE with centroid G Prove: EG = 2/3EB Statement Centroid Theorem Proof [Centroid Thm Prf] Reason 1. △AEC with centroid G 1. Given 3. Ⓐ△AGE = Ⓐ△EGF + Ⓐ△AGF 3. Partition Postulate 2. Ⓐ△AGB = Ⓐ△AGF = Ⓐ△EGF 4. Ⓐ△AGE = Ⓐ△AGB + Ⓐ△AGB 5. Ⓐ△AGE = 2( Ⓐ△AGB ) 6 . Let h be the common height of △AGE and △AGB 7. Ⓐ△AGB = 1/2(GB)h 8. Ⓐ△AGE = 2(1/2(GB)h) 9. Ⓐ△AGE = (GB)h 10. Ⓐ△AGE = 1/2(EG)h 2. ⒶCentroid△s 4. Substitution Prop 5. Combine Terms 6. Substitution Prop 7. △ Area 8. Substitution Prop 9. Product 10. △ Area 11. (GB)h = 1/2(EG)h 11. Transitive Prop 12. (GB)h/h = 1/2(EG)h/h 12. Division Prop 13. GB = 1/2EG 13. Quotient 14. 2GB = 2(1/2EG) 14. Substitution Prop 15. 2GB = EG 15. Product 16. EB = EG + GB 16. Segment Addition 17. EB = 2GB + GB 17. Substitution Prop 18. EB = 3GB 18. Combine Terms 19. EG = 2GB 19. Symmetric Prop 20. EG/EB = 2GB/EB 20. Division Prop next page Given: △ACE with centroid G Prove: EG = 2/3EB Statement Centroid Theorem Proof [Centroid Thm Prf] Reason 21. (EG/EB)EB = 2/3EB 1. Multiplication Prop 2. EG = 2/3 EB 2. Product Given: AX bisects TI; IZ bisects AE; XI ≊ ZE; AX and IZ trisect TE; right TIN is a EAR is a right Prove: IN ≊ AR Statement Guitar Strings Proof [ Guitar Strs Prf] Reason 1. AX bisects TI 1. Given 2. X is the midpoint of TI 2. Segment Bisector 3. 2(XI) = TI 3. Segment Midpoint 4. 2(XI)/2 = TI/2 4. Division Prop 5. XI = TI/2 5. Quotient 6. IZ bisects AE 6. Given 7. Z is the midpoint of AE 7. Segment Bisector 8. 2(ZE) = AE 8. Segment Midpoint 9. 2(ZE)/2 = AE/2 9. Division Prop 10. ZE = AE/2 10. Quotient 11. XI ≊ ZE 11. Given 12. XI = ZE 12. Segs ≊ IFF Lens= 13. TI/2 = AE/2 13. Substitution Prop 14. 2(TI/2) = 2(AE/2) 14. Multiplication Prop 15. TI = AE 15. Product 16. TI ≊ AE 16. Segs ≊ IFF Lens= 17. AX and IZ trisect TE 17. Given 18. TR ≊ RN ≊ NE 18. Segment Trisector 19. TR = RN = NE 19. Segs ≊ IFF Lens= 20. TR + RN = RN + NE 20. Segment Addition next page Given: AX bisects TI; IZ bisects AE; AX and IZ trisect TE; XI ≊ ZE; TIN is a right EAR is a right Prove: IN ≊ AR Guitar Strings Proof [ Guitar Strs Prf] Statement Reason 21. TR + RN = TN 21. Segment Addtion 22. RN + NE = RE 22. Segment Addtion 23. TN = RE 23. Transitive Prop 24. TN ≊ RE 24. Segs ≊ IFF Lens= 25. ⊀TIN is a right ⊀ 25. Given 26. ⊀EAR is a right ⊀ 26. Given 28. IN ≊ AR 28. Corr Pts ≊ △s ≊ 27. △TIN ≊ △EAR 27. HL Right △s ≊ Given: △LMN is isosceles; O is the midpoint of LM; P is the midpoint of NM; OQ ⊥ LN; PR ⊥ LN Prove: OQ ≊ PR Ski Chalet Proof Statement 1. △LMN is isosceles [Ski Chalet Prf] Reason 1. Given 2. LM ≊ MN 2. Isosceles △ 3. LM = MN 3. Segs ≊ IFF Lens= 4. L ≊ N 4. Isos△ SidesIFF s 5. O is the midpoint of LM 5. Given 6. LO ≊ OM 6. Segment Midpoint 7. P is the midpoint of NM 7. Given 8. NP ≊ PM 8. Segment Midpoint 9. LM = LO + OM 9. Segment Addition 10. NM = NP + PM 10. Segment Addition 11. LO + OM = NP + PM 11. Transitive Prop 12. LO + LO = NP + NP 12. Substution Prop 13. 2LO = 2NP 13. Combine Terms 14. 2LO/2 = 2NP/2 14. Division Prop 15. LO = NP 15. Quotient 16. LO ≊ NP 16. Segs ≊ IFF Lens= 17. OQ ⊥ LN 17. Given 18. PR ⊥ LN 18. Given 19. OQL is a right angle 19. ⊥ Lines 4Rt 20. PRN is a right angle 20. ⊥ Lines 4Rt next page s s Given: △LMN is isosceles; O is the midpoint of LM; P is the midpoint of MN; DQ ⊥ LN; PR ⊥ LN Prove: OQ ≊ PR Statement Ski Chalet Proof 21. OQL is a right triangle 22. PRN is a right triangle 23. OQL ≊ 24. OQ ≊ PR PRN [Ski Chalet Prf] Reason 21. Right △ 22. Right △ 23. HA Right △ ≊ 24. Corr Pts ≊ △s Given: AB || DC; △AED ≊ △CFB Prove: △ABE ≊ △CDF Boxed Triangles Proof Statement [Boxed △s Prf] 1. △AED ≊ △CFB 2. 1 ≊ 3 3. 5 ≊ 6 Reason 1. Given 2. Corr Pts ≊ △s 3. Corr Pts ≊ △s 4. DE ≊ BF 4. Corr Pts ≊ △s 5. DE = BF 5. Segs ≊ IFF Lens= 6. AB || DC 6. Given 7. 7. Alternate Int 7 ≊ 8 s 8. EF = FE 8. Reflexive Prop 9. DE + EF = BF + FE 9. Segment Addition 10. DE + EF = DF 10. Segment Addition 11. BF + FE = BE 11. Segment Addition 12. DF = BE 12. Transitive Prop 13. 13. A=m 1+m 2 Addition 14. m 6 + m A + m 7 = 180° 14. Sum 15. 15. C= m 3 + m 4 s Of △ Addition s Of △ 16. m 5 + m C + m 8 = 180° 16. Sum 17. m 6 + m B + m 7 = m 5 + m C + m 8 17. Transitive Prop 18. m 6 + m 1 + m 2 + m 7 = m 5 + m 3 + m 4 + m 8 18. Substitution Prop 19. m 6 + m 1 + m 2 + m 7 - m 1 = m 5 + m 1 + m 4 + m 8-m 1 19. Subtraction Prop 20. m 6 + m 2 + m 7 = m 5 + m 4 + m 8 20. Combine Terms next page Given: AB || DC; △AED ≊ △CFB Prove: △ABE ≊ △CDF Statement Boxed Triangles Proof 21. m 5 + m 2 + m 7 = m 5 + m 4 + m 8 [Boxed △s Prf] Reason 21. Substitution Prop 22. m 5 + m 2 + m 7 - m 5 = m 5 + m 4 + m 8 - m 5 22. Subtraction Prop 23. m 2 + m 7 = m 4 + m 8 23. Combine Terms 24. m 2 + m 7 = m 4 + m 7 24. Substution Prop 25. m 2 + m 7 - m 7 = m 4 + m 7 - m 7 25. SubtractionProp 26.m 2 = m 4 26. Combine Terms 27. △ABE ≊ △CDF ⁂ 27. AAS △ ≊ Given: △ ABC is equilateral; P is the centroid of △ABC Prove: △AFP ≊ △ADP ≊ BDP ≊ △BEP ≊ △CEP ≊ △CFP Equilateral Centroid Triangles Proof Statement [EquLat Centroid Prf] Reason 1. P is the centroid of △ABC 1. Given 3. D is the midpoint of AB 3. △ Median 4. AD = DB = 1/2AB 4. Midpoint Thm 6. F is the midpoint of AC 6. △ Median 7. AF = FC = 1/2AC 7. Midpoint Thm 9. E is the midpoint of BC 9. △ Median 10. CE = BE = 1/2BC 10. Midpoint Thm 2. CD is a median of △ABC 5. BF is a median of △ABC 8. AE is a median of △ABC 11. △ABC is equilateral 2. △ Centroid 5. △ Centroid 8. △ Centroid 11. Given 12. AB ≊ AC ≊ BC 12. Equilateral 13. AB = AC = BC 13. Segs ≊ IFF = Length 14. 1/2AB = 1/2AC = 1/2CB 14. Mult Prop 15. AD = DB = AF = FC = CE = BE 15. Transitive Prop 16. AD ≊ DB ≊ AF ≊ FC ≊ CE ≊ BE 16. Segs ≊ IFF = Length 17. AP ≊ AP 17. Reflexive Prop ≊ 19. DP ≊ DP 19. Reflexive Prop ≊ 18. △AFP ≊ △ADP 20. △ADP ≊ △BDP next page 18. SSS △ ≊ 20. SSS △ ≊ Given: △ ABC is equilateral; P is the centroid of △ABC Prove: △AFP ≊ △ADP ≊ BDP ≊ △BEP ≊ △CEP ≊ △CFP Equilateral Centroid Proof [EquLat Centroid Prf] Statement Reason 21. PB ≊ PB 21. Reflexive Prop ≊ 23. EP ≊ EP 23. Reflexive Prop ≊ 25. CP ≊ CP 25. Reflexive Prop ≊ 27. FP ≊ FP 27. Reflexive Prop ≊ 22. △BDP ≊ △BEP 24. △BEP ≊ △CEP 26. △CEP ≊ △CFP 28. △CFP ≊ △AFP 29. △AFP ≊ △ADP ≊ △DBP ≊ △BEP ≊ △CEP ≊ △CFP ⁂ 22. SSS △ ≊ 24. SSS △ ≊ 26. SSS △ ≊ 28. SSS △ ≊ 29. Transitive Prop ≊ Given: m A = 30°; B is trisected by BM and BT; C is trisected by CM and CT Prove: m M = 80°; m T = 130° Russian Dolls Proof [△ Trisection Prf] Statement Reason 1. B is trisected by BM and BT 1. Given 2. ABM 2. MBT TBC Trisector 3. m ABM MBT TBC 3. Segs IFF=Len 4. m ABM MBT TBC = m B 4. Partition Postulate 5. m ABM ABM ABM = m B 5. Substitution Prop 6. 3m ABM = m B 6. Combine Terms 7. (1/3)3m ABM = (1/3)m B 7. Multiplication Prop 8. m ABM = (1/3)m B 8. Mult Inverse 9. m MBT = m TBC = (1/3)m B 9. Given 10. 10. 11. C is trisected by CM and CT ACM MCT TCB Trisector 11. Segs IFF=Len 12. m ACM MCT TCB 12. Partition Postulate 13. m ACM MCT TCB = m C 13. Substitution Prop 14. m ACM ACM ACM = m C 14. Combine Terms 15. 3m ACM = m C 15. Multiplication Prop 16. (1/3)3m ACM = (1/3)m C 16. Mult Inverse 17. m ACM = (1/3)m C 17. 18. m MCT = m TCB = (1/3)m C 18. Substitution Prop 19. m A = 30° 19. Distributive Prop 20. m A + m ABC + m ACB = 180 20. Given Addition Given: m A = 30°; ABC is trisected by BM and BT; ACB is trisected by CM and CT Prove: m Statement M = 80°; m T = 130° Russian Dolls Proof [Russian Dolls Prf] Reason 21. 30° + m ABC + m ACB = 180° 21. Substitution Prop 22. 30° + m ABC + m ACB - 30° = 180° - 30° 22. Substitution Prop 23. m ABC + m ACB = 150° 23. Combine Terms 24. m MBC = m MBT + m TBC 24. 25. m MBC = (1/3)m ABC +(1/3) m ABC 25. Substitution Prop 26. m MBC = (2/3)m ABC 26. Distributive Prop 27. m MCB = m MCT + m TCB 27. Substitution Prop 28. m MCB = (1/3)m ACB +(1/3) m ACB 28. Product 29. m MCB = (2/3)m ACB 29. Subtraction Prop 30. m M + m MBC + m 30. Combine Terms 31. m M + (2/3)m 32. m M + (2/3)(m MCB = 180° ABC + (2/3)m ACB = 180 ABC + m ACB) = 180° Addition 31. Sum s Of △ 32. Substitution Prop 33. m M + (2/3)(150°) = 180° 33. Distributive Prop 34. m M + 100° = 180° 34. Substitution Prop 35. m M + 100° - 100°= 180° - 100° 35. Product 36. m M = 80° 36. Subtraction Prop 37. m T + m CBT + m BCT = 180° 37. CombineTerms 38. m T + 1/3(m ABC) + (1/3)m ACB = 180° 38. Substitution Prop 39. m T + (1/3)(m ABC + m ACB) = 180° 39. Distributive Prop 40. m T + (1/3)(150°) = 180° 40. Substitution Prop next page Given: m A = 30°; ABC is trisected by BM and BT; ACB is trisected by CM and CT Prove: m Statement M = 80°; m T = 130° Russian Dolls Proof [Russian Dolls Prf] Reason 41. m T + 50° = 180° 21. Product 42. m T + 50° - 50° = 180° - 50° 22. Subtraction Prop 43. m T = 130° 23. Combine Terms
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