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Proof: The Isosceles Triangle Base Angles Theorem (Geometry)

Free printable Geometry geometry worksheet: complete a two-column proof that the base angles of an isosceles triangle are congruent, using SAS and CPCTC.

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Geometry Proof: The Isosceles Triangle Base Angles Theorem

Complete each two-column proof that the base angles of an isosceles triangle are congruent, using the angle-bisector construction, SAS and CPCTC.

  1. 1.
    Triangle XYZ with XY ≅\cong XZ. Prove: ∠Y\angle Y ≅\cong ∠Z\angle Z. Complete the missing statement in step 1 of the two-column proof.
    XYZW
    StatementReason
    1. Given
    2.Draw XW, the bisector of ∠X\angle X, meeting YZ at W.Construction (every angle has exactly one bisector)
    3.∠YXW\angle YXW ≅\cong ∠ZXW\angle ZXW.Definition of an angle bisector
    4.XW ≅\cong XW.Reflexive Property of Congruence
    5.△YXW\triangle YXW ≅\cong △ZXW\triangle ZXW.SAS (Side-Angle-Side) Congruence
    6.∠Y\angle Y ≅\cong ∠Z\angle Z.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  2. 2.
    Triangle PQR with PQ ≅\cong PR. Prove: ∠Q\angle Q ≅\cong ∠R\angle R. Complete the missing statement in step 3 of the two-column proof.
    PQRS
    StatementReason
    1.PQ ≅\cong PR.Given
    2.Draw PS, the bisector of ∠P\angle P, meeting QR at S.Construction (every angle has exactly one bisector)
    3. Definition of an angle bisector
    4.PS ≅\cong PS.Reflexive Property of Congruence
    5.△QPS\triangle QPS ≅\cong △RPS\triangle RPS.SAS (Side-Angle-Side) Congruence
    6.∠Q\angle Q ≅\cong ∠R\angle R.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  3. 3.
    Triangle XYZ with XY ≅\cong XZ. Prove: ∠Y\angle Y ≅\cong ∠Z\angle Z. Complete the missing reason in step 6 of the two-column proof.
    XYZW
    StatementReason
    1.XY ≅\cong XZ.Given
    2.Draw XW, the bisector of ∠X\angle X, meeting YZ at W.Construction (every angle has exactly one bisector)
    3.∠YXW\angle YXW ≅\cong ∠ZXW\angle ZXW.Definition of an angle bisector
    4.XW ≅\cong XW.Reflexive Property of Congruence
    5.△YXW\triangle YXW ≅\cong △ZXW\triangle ZXW.SAS (Side-Angle-Side) Congruence
    6.∠Y\angle Y ≅\cong ∠Z\angle Z. 
  4. 4.
    Triangle EFG with EF ≅\cong EG. Prove: ∠F\angle F ≅\cong ∠G\angle G. Complete the missing statement in step 4 of the two-column proof.
    EFGH
    StatementReason
    1.EF ≅\cong EG.Given
    2.Draw EH, the bisector of ∠E\angle E, meeting FG at H.Construction (every angle has exactly one bisector)
    3.∠FEH\angle FEH ≅\cong ∠GEH\angle GEH.Definition of an angle bisector
    4. Reflexive Property of Congruence
    5.△FEH\triangle FEH ≅\cong △GEH\triangle GEH.SAS (Side-Angle-Side) Congruence
    6.∠F\angle F ≅\cong ∠G\angle G.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  5. 5.
    Triangle ABC with AB ≅\cong AC. Prove: ∠B\angle B ≅\cong ∠C\angle C. Complete the missing statement in step 5 of the two-column proof.
    ABCD
    StatementReason
    1.AB ≅\cong AC.Given
    2.Draw AD, the bisector of ∠A\angle A, meeting BC at D.Construction (every angle has exactly one bisector)
    3.∠BAD\angle BAD ≅\cong ∠CAD\angle CAD.Definition of an angle bisector
    4.AD ≅\cong AD.Reflexive Property of Congruence
    5. SAS (Side-Angle-Side) Congruence
    6.∠B\angle B ≅\cong ∠C\angle C.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  6. 6.
    Triangle JKL with JK ≅\cong JL. Prove: ∠K\angle K ≅\cong ∠L\angle L. Complete the missing statement in step 6 of the two-column proof.
    JKLM
    StatementReason
    1.JK ≅\cong JL.Given
    2.Draw JM, the bisector of ∠J\angle J, meeting KL at M.Construction (every angle has exactly one bisector)
    3.∠KJM\angle KJM ≅\cong ∠LJM\angle LJM.Definition of an angle bisector
    4.JM ≅\cong JM.Reflexive Property of Congruence
    5.△KJM\triangle KJM ≅\cong △LJM\triangle LJM.SAS (Side-Angle-Side) Congruence
    6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  7. 7.
    Triangle JKL with JK ≅\cong JL. Prove: ∠K\angle K ≅\cong ∠L\angle L. Complete the missing statement in step 2 of the two-column proof.
    JKLM
    StatementReason
    1.JK ≅\cong JL.Given
    2. Construction (every angle has exactly one bisector)
    3.∠KJM\angle KJM ≅\cong ∠LJM\angle LJM.Definition of an angle bisector
    4.JM ≅\cong JM.Reflexive Property of Congruence
    5.△KJM\triangle KJM ≅\cong △LJM\triangle LJM.SAS (Side-Angle-Side) Congruence
    6.∠K\angle K ≅\cong ∠L\angle L.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  8. 8.
    Triangle JKL with JK ≅\cong JL. Prove: ∠K\angle K ≅\cong ∠L\angle L. Complete the missing reason in step 1 of the two-column proof.
    JKLM
    StatementReason
    1.JK ≅\cong JL. 
    2.Draw JM, the bisector of ∠J\angle J, meeting KL at M.Construction (every angle has exactly one bisector)
    3.∠KJM\angle KJM ≅\cong ∠LJM\angle LJM.Definition of an angle bisector
    4.JM ≅\cong JM.Reflexive Property of Congruence
    5.△KJM\triangle KJM ≅\cong △LJM\triangle LJM.SAS (Side-Angle-Side) Congruence
    6.∠K\angle K ≅\cong ∠L\angle L.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  9. 9.
    Triangle ABC with AB ≅\cong AC. Prove: ∠B\angle B ≅\cong ∠C\angle C. Complete the missing statement in step 1 of the two-column proof.
    ABCD
    StatementReason
    1. Given
    2.Draw AD, the bisector of ∠A\angle A, meeting BC at D.Construction (every angle has exactly one bisector)
    3.∠BAD\angle BAD ≅\cong ∠CAD\angle CAD.Definition of an angle bisector
    4.AD ≅\cong AD.Reflexive Property of Congruence
    5.△BAD\triangle BAD ≅\cong △CAD\triangle CAD.SAS (Side-Angle-Side) Congruence
    6.∠B\angle B ≅\cong ∠C\angle C.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  10. 10.
    Triangle ABC with AB ≅\cong AC. Prove: ∠B\angle B ≅\cong ∠C\angle C. Complete the missing reason in step 1 of the two-column proof.
    ABCD
    StatementReason
    1.AB ≅\cong AC. 
    2.Draw AD, the bisector of ∠A\angle A, meeting BC at D.Construction (every angle has exactly one bisector)
    3.∠BAD\angle BAD ≅\cong ∠CAD\angle CAD.Definition of an angle bisector
    4.AD ≅\cong AD.Reflexive Property of Congruence
    5.△BAD\triangle BAD ≅\cong △CAD\triangle CAD.SAS (Side-Angle-Side) Congruence
    6.∠B\angle B ≅\cong ∠C\angle C.CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
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