ChalkBee
Make

Proof: The Triangle Angle Sum Theorem (Geometry)

Free printable Geometry geometry worksheet: complete a two-column proof that a triangle's interior angles sum to 180°, using a constructed parallel line.

✓ Answer key checked by math, never wrong
Geometry · Math worksheet
Name
Date
Math

Geometry Proof: The Triangle Angle Sum Theorem

Complete each two-column proof that the interior angles of a triangle sum to 180°, using a constructed parallel line and the Alternate Interior Angles Theorem.

  1. 1.
    Triangle RPQ. Prove: m∠Rm\angle R + m∠Pm\angle P + m∠Qm\angle Q = 180°. Complete the missing reason in step 5 of the two-column proof.
    RPQn∠1∠2
    StatementReason
    1.△RPQ\triangle RPQ is a triangle.Given
    2.Draw line n through Q, parallel to line RP.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠R\angle R and ∠2\angle 2 ≅\cong ∠P\angle P (∠1\angle 1 and ∠2\angle 2 are the angles n makes with QR and QP, on either side of ∠Q\angle Q).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Rm\angle R and m∠2m\angle 2 = m∠Pm\angle P.Definition of congruent angles
    5.m∠1m\angle 1 + m∠RQPm\angle RQP + m∠2m\angle 2 = 180°. 
    6.m∠Rm\angle R + m∠Qm\angle Q + m∠Pm\angle P = 180°.Substitution Property of Equality
    7.m∠Rm\angle R + m∠Pm\angle P + m∠Qm\angle Q = 180°.Commutative Property of Addition
  2. 2.
    Triangle EFD. Prove: m∠Em\angle E + m∠Fm\angle F + m∠Dm\angle D = 180°. Complete the missing statement in step 7 of the two-column proof.
    EFDn∠1∠2
    StatementReason
    1.△EFD\triangle EFD is a triangle.Given
    2.Draw line n through D, parallel to line EF.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠E\angle E and ∠2\angle 2 ≅\cong ∠F\angle F (∠1\angle 1 and ∠2\angle 2 are the angles n makes with DE and DF, on either side of ∠D\angle D).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Em\angle E and m∠2m\angle 2 = m∠Fm\angle F.Definition of congruent angles
    5.m∠1m\angle 1 + m∠EDFm\angle EDF + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Em\angle E + m∠Dm\angle D + m∠Fm\angle F = 180°.Substitution Property of Equality
    7. Commutative Property of Addition
  3. 3.
    Triangle TUS. Prove: m∠Tm\angle T + m∠Um\angle U + m∠Sm\angle S = 180°. Complete the missing statement in step 4 of the two-column proof.
    TUSn∠1∠2
    StatementReason
    1.△TUS\triangle TUS is a triangle.Given
    2.Draw line n through S, parallel to line TU.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠T\angle T and ∠2\angle 2 ≅\cong ∠U\angle U (∠1\angle 1 and ∠2\angle 2 are the angles n makes with ST and SU, on either side of ∠S\angle S).Alternate Interior Angles Theorem
    4. Definition of congruent angles
    5.m∠1m\angle 1 + m∠TSUm\angle TSU + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Tm\angle T + m∠Sm\angle S + m∠Um\angle U = 180°.Substitution Property of Equality
    7.m∠Tm\angle T + m∠Um\angle U + m∠Sm\angle S = 180°.Commutative Property of Addition
  4. 4.
    Triangle FDE. Prove: m∠Fm\angle F + m∠Dm\angle D + m∠Em\angle E = 180°. Complete the missing statement in step 4 of the two-column proof.
    FDEn∠1∠2
    StatementReason
    1.△FDE\triangle FDE is a triangle.Given
    2.Draw line n through E, parallel to line FD.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠F\angle F and ∠2\angle 2 ≅\cong ∠D\angle D (∠1\angle 1 and ∠2\angle 2 are the angles n makes with EF and ED, on either side of ∠E\angle E).Alternate Interior Angles Theorem
    4. Definition of congruent angles
    5.m∠1m\angle 1 + m∠FEDm\angle FED + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Fm\angle F + m∠Em\angle E + m∠Dm\angle D = 180°.Substitution Property of Equality
    7.m∠Fm\angle F + m∠Dm\angle D + m∠Em\angle E = 180°.Commutative Property of Addition
  5. 5.
    Triangle LMK. Prove: m∠Lm\angle L + m∠Mm\angle M + m∠Km\angle K = 180°. Complete the missing statement in step 1 of the two-column proof.
    LMKn∠1∠2
    StatementReason
    1. Given
    2.Draw line n through K, parallel to line LM.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠L\angle L and ∠2\angle 2 ≅\cong ∠M\angle M (∠1\angle 1 and ∠2\angle 2 are the angles n makes with KL and KM, on either side of ∠K\angle K).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Lm\angle L and m∠2m\angle 2 = m∠Mm\angle M.Definition of congruent angles
    5.m∠1m\angle 1 + m∠LKMm\angle LKM + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Lm\angle L + m∠Km\angle K + m∠Mm\angle M = 180°.Substitution Property of Equality
    7.m∠Lm\angle L + m∠Mm\angle M + m∠Km\angle K = 180°.Commutative Property of Addition
  6. 6.
    Triangle DEF. Prove: m∠Dm\angle D + m∠Em\angle E + m∠Fm\angle F = 180°. Complete the missing statement in step 1 of the two-column proof.
    DEFn∠1∠2
    StatementReason
    1. Given
    2.Draw line n through F, parallel to line DE.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠D\angle D and ∠2\angle 2 ≅\cong ∠E\angle E (∠1\angle 1 and ∠2\angle 2 are the angles n makes with FD and FE, on either side of ∠F\angle F).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Dm\angle D and m∠2m\angle 2 = m∠Em\angle E.Definition of congruent angles
    5.m∠1m\angle 1 + m∠DFEm\angle DFE + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Dm\angle D + m∠Fm\angle F + m∠Em\angle E = 180°.Substitution Property of Equality
    7.m∠Dm\angle D + m∠Em\angle E + m∠Fm\angle F = 180°.Commutative Property of Addition
  7. 7.
    Triangle TUS. Prove: m∠Tm\angle T + m∠Um\angle U + m∠Sm\angle S = 180°. Complete the missing statement in step 6 of the two-column proof.
    TUSn∠1∠2
    StatementReason
    1.△TUS\triangle TUS is a triangle.Given
    2.Draw line n through S, parallel to line TU.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠T\angle T and ∠2\angle 2 ≅\cong ∠U\angle U (∠1\angle 1 and ∠2\angle 2 are the angles n makes with ST and SU, on either side of ∠S\angle S).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Tm\angle T and m∠2m\angle 2 = m∠Um\angle U.Definition of congruent angles
    5.m∠1m\angle 1 + m∠TSUm\angle TSU + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6. Substitution Property of Equality
    7.m∠Tm\angle T + m∠Um\angle U + m∠Sm\angle S = 180°.Commutative Property of Addition
  8. 8.
    Triangle STU. Prove: m∠Sm\angle S + m∠Tm\angle T + m∠Um\angle U = 180°. Complete the missing statement in step 4 of the two-column proof.
    STUn∠1∠2
    StatementReason
    1.△STU\triangle STU is a triangle.Given
    2.Draw line n through U, parallel to line ST.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠S\angle S and ∠2\angle 2 ≅\cong ∠T\angle T (∠1\angle 1 and ∠2\angle 2 are the angles n makes with US and UT, on either side of ∠U\angle U).Alternate Interior Angles Theorem
    4. Definition of congruent angles
    5.m∠1m\angle 1 + m∠SUTm\angle SUT + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Sm\angle S + m∠Um\angle U + m∠Tm\angle T = 180°.Substitution Property of Equality
    7.m∠Sm\angle S + m∠Tm\angle T + m∠Um\angle U = 180°.Commutative Property of Addition
  9. 9.
    Triangle RPQ. Prove: m∠Rm\angle R + m∠Pm\angle P + m∠Qm\angle Q = 180°. Complete the missing statement in step 6 of the two-column proof.
    RPQn∠1∠2
    StatementReason
    1.△RPQ\triangle RPQ is a triangle.Given
    2.Draw line n through Q, parallel to line RP.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠R\angle R and ∠2\angle 2 ≅\cong ∠P\angle P (∠1\angle 1 and ∠2\angle 2 are the angles n makes with QR and QP, on either side of ∠Q\angle Q).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Rm\angle R and m∠2m\angle 2 = m∠Pm\angle P.Definition of congruent angles
    5.m∠1m\angle 1 + m∠RQPm\angle RQP + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6. Substitution Property of Equality
    7.m∠Rm\angle R + m∠Pm\angle P + m∠Qm\angle Q = 180°.Commutative Property of Addition
  10. 10.
    Triangle XYZ. Prove: m∠Xm\angle X + m∠Ym\angle Y + m∠Zm\angle Z = 180°. Complete the missing statement in step 1 of the two-column proof.
    XYZn∠1∠2
    StatementReason
    1. Given
    2.Draw line n through Z, parallel to line XY.Construction (Parallel Postulate)
    3.∠1\angle 1 ≅\cong ∠X\angle X and ∠2\angle 2 ≅\cong ∠Y\angle Y (∠1\angle 1 and ∠2\angle 2 are the angles n makes with ZX and ZY, on either side of ∠Z\angle Z).Alternate Interior Angles Theorem
    4.m∠1m\angle 1 = m∠Xm\angle X and m∠2m\angle 2 = m∠Ym\angle Y.Definition of congruent angles
    5.m∠1m\angle 1 + m∠XZYm\angle XZY + m∠2m\angle 2 = 180°.Angles on a straight line sum to 180°
    6.m∠Xm\angle X + m∠Zm\angle Z + m∠Ym\angle Y = 180°.Substitution Property of Equality
    7.m∠Xm\angle X + m∠Ym\angle Y + m∠Zm\angle Z = 180°.Commutative Property of Addition
Made with ChalkBee · chalkbee.com

More like this

Vertical angles proofGeometry · printable PDFParallel lines proofGeometry · printable PDFIsosceles base angles proofGeometry · printable PDFAngle theorem applicationsGeometry · printable PDFDigraphsKindergarten · printable PDFBlendsGrade 1 · printable PDF