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Proof: The Vertical Angles Theorem (Geometry)

Free printable Geometry geometry worksheet: complete a two-column proof that vertical angles are congruent, using the Linear Pair Postulate.

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Geometry ¡ Math worksheet
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Geometry Proof: The Vertical Angles Theorem

Two lines intersect, forming two pairs of vertical angles. Complete each two-column proof that the vertical angles are congruent by filling in the missing statement or reason.

  1. 1.
    Two lines intersect at O, with B, O, D collinear and C, O, A collinear. Prove: ∠BOC ≅ ∠DOA. Complete the missing statement in step 3 of the two-column proof.
    BCDAO
    StatementReason
    1.B, O, D are collinear, and C, O, A are collinear.Given
    2.∠BOC and ∠COD form a linear pair.Definition of a linear pair (BOD is a straight line)
    3. Definition of a linear pair (COA is a straight line)
    4.m∠BOC + m∠COD = 180°, and m∠COD + m∠DOA = 180°.Linear Pair Postulate
    5.m∠BOC + m∠COD = m∠COD + m∠DOA.Transitive Property of Equality
    6.m∠BOC = m∠DOA.Subtraction Property of Equality
    7.∠BOC ≅ ∠DOA.Definition of congruent angles
  2. 2.
    Two lines intersect at P, with Q, P, S collinear and R, P, T collinear. Prove: ∠QPR ≅ ∠SPT. Complete the missing statement in step 4 of the two-column proof.
    QRSTP
    StatementReason
    1.Q, P, S are collinear, and R, P, T are collinear.Given
    2.∠QPR and ∠RPS form a linear pair.Definition of a linear pair (QPS is a straight line)
    3.∠RPS and ∠SPT form a linear pair.Definition of a linear pair (RPT is a straight line)
    4. Linear Pair Postulate
    5.m∠QPR + m∠RPS = m∠RPS + m∠SPT.Transitive Property of Equality
    6.m∠QPR = m∠SPT.Subtraction Property of Equality
    7.∠QPR ≅ ∠SPT.Definition of congruent angles
  3. 3.
    Two lines intersect at O, with A, O, C collinear and B, O, D collinear. Prove: ∠AOB ≅ ∠COD. Complete the missing reason in step 2 of the two-column proof.
    ABCDO
    StatementReason
    1.A, O, C are collinear, and B, O, D are collinear.Given
    2.∠AOB and ∠BOC form a linear pair. 
    3.∠BOC and ∠COD form a linear pair.Definition of a linear pair (BOD is a straight line)
    4.m∠AOB + m∠BOC = 180°, and m∠BOC + m∠COD = 180°.Linear Pair Postulate
    5.m∠AOB + m∠BOC = m∠BOC + m∠COD.Transitive Property of Equality
    6.m∠AOB = m∠COD.Subtraction Property of Equality
    7.∠AOB ≅ ∠COD.Definition of congruent angles
  4. 4.
    Two lines intersect at K, with E, K, G collinear and F, K, H collinear. Prove: ∠EKF ≅ ∠GKH. Complete the missing reason in step 3 of the two-column proof.
    EFGHK
    StatementReason
    1.E, K, G are collinear, and F, K, H are collinear.Given
    2.∠EKF and ∠FKG form a linear pair.Definition of a linear pair (EKG is a straight line)
    3.∠FKG and ∠GKH form a linear pair. 
    4.m∠EKF + m∠FKG = 180°, and m∠FKG + m∠GKH = 180°.Linear Pair Postulate
    5.m∠EKF + m∠FKG = m∠FKG + m∠GKH.Transitive Property of Equality
    6.m∠EKF = m∠GKH.Subtraction Property of Equality
    7.∠EKF ≅ ∠GKH.Definition of congruent angles
  5. 5.
    Two lines intersect at N, with L, N, V collinear and U, N, J collinear. Prove: ∠LNU ≅ ∠VNJ. Complete the missing reason in step 2 of the two-column proof.
    LUVJN
    StatementReason
    1.L, N, V are collinear, and U, N, J are collinear.Given
    2.∠LNU and ∠UNV form a linear pair. 
    3.∠UNV and ∠VNJ form a linear pair.Definition of a linear pair (UNJ is a straight line)
    4.m∠LNU + m∠UNV = 180°, and m∠UNV + m∠VNJ = 180°.Linear Pair Postulate
    5.m∠LNU + m∠UNV = m∠UNV + m∠VNJ.Transitive Property of Equality
    6.m∠LNU = m∠VNJ.Subtraction Property of Equality
    7.∠LNU ≅ ∠VNJ.Definition of congruent angles
  6. 6.
    Two lines intersect at N, with J, N, U collinear and L, N, V collinear. Prove: ∠JNL ≅ ∠UNV. Complete the missing statement in step 3 of the two-column proof.
    JLUVN
    StatementReason
    1.J, N, U are collinear, and L, N, V are collinear.Given
    2.∠JNL and ∠LNU form a linear pair.Definition of a linear pair (JNU is a straight line)
    3. Definition of a linear pair (LNV is a straight line)
    4.m∠JNL + m∠LNU = 180°, and m∠LNU + m∠UNV = 180°.Linear Pair Postulate
    5.m∠JNL + m∠LNU = m∠LNU + m∠UNV.Transitive Property of Equality
    6.m∠JNL = m∠UNV.Subtraction Property of Equality
    7.∠JNL ≅ ∠UNV.Definition of congruent angles
  7. 7.
    Two lines intersect at M, with X, M, Z collinear and Y, M, W collinear. Prove: ∠XMY ≅ ∠ZMW. Complete the missing reason in step 5 of the two-column proof.
    XYZWM
    StatementReason
    1.X, M, Z are collinear, and Y, M, W are collinear.Given
    2.∠XMY and ∠YMZ form a linear pair.Definition of a linear pair (XMZ is a straight line)
    3.∠YMZ and ∠ZMW form a linear pair.Definition of a linear pair (YMW is a straight line)
    4.m∠XMY + m∠YMZ = 180°, and m∠YMZ + m∠ZMW = 180°.Linear Pair Postulate
    5.m∠XMY + m∠YMZ = m∠YMZ + m∠ZMW. 
    6.m∠XMY = m∠ZMW.Subtraction Property of Equality
    7.∠XMY ≅ ∠ZMW.Definition of congruent angles
  8. 8.
    Two lines intersect at N, with J, N, U collinear and L, N, V collinear. Prove: ∠JNL ≅ ∠UNV. Complete the missing reason in step 5 of the two-column proof.
    JLUVN
    StatementReason
    1.J, N, U are collinear, and L, N, V are collinear.Given
    2.∠JNL and ∠LNU form a linear pair.Definition of a linear pair (JNU is a straight line)
    3.∠LNU and ∠UNV form a linear pair.Definition of a linear pair (LNV is a straight line)
    4.m∠JNL + m∠LNU = 180°, and m∠LNU + m∠UNV = 180°.Linear Pair Postulate
    5.m∠JNL + m∠LNU = m∠LNU + m∠UNV. 
    6.m∠JNL = m∠UNV.Subtraction Property of Equality
    7.∠JNL ≅ ∠UNV.Definition of congruent angles
  9. 9.
    Two lines intersect at K, with E, K, G collinear and F, K, H collinear. Prove: ∠EKF ≅ ∠GKH. Complete the missing statement in step 3 of the two-column proof.
    EFGHK
    StatementReason
    1.E, K, G are collinear, and F, K, H are collinear.Given
    2.∠EKF and ∠FKG form a linear pair.Definition of a linear pair (EKG is a straight line)
    3. Definition of a linear pair (FKH is a straight line)
    4.m∠EKF + m∠FKG = 180°, and m∠FKG + m∠GKH = 180°.Linear Pair Postulate
    5.m∠EKF + m∠FKG = m∠FKG + m∠GKH.Transitive Property of Equality
    6.m∠EKF = m∠GKH.Subtraction Property of Equality
    7.∠EKF ≅ ∠GKH.Definition of congruent angles
  10. 10.
    Two lines intersect at K, with E, K, G collinear and F, K, H collinear. Prove: ∠EKF ≅ ∠GKH. Complete the missing statement in step 2 of the two-column proof.
    EFGHK
    StatementReason
    1.E, K, G are collinear, and F, K, H are collinear.Given
    2. Definition of a linear pair (EKG is a straight line)
    3.∠FKG and ∠GKH form a linear pair.Definition of a linear pair (FKH is a straight line)
    4.m∠EKF + m∠FKG = 180°, and m∠FKG + m∠GKH = 180°.Linear Pair Postulate
    5.m∠EKF + m∠FKG = m∠FKG + m∠GKH.Transitive Property of Equality
    6.m∠EKF = m∠GKH.Subtraction Property of Equality
    7.∠EKF ≅ ∠GKH.Definition of congruent angles
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