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.
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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.Triangle RPQ. Prove: m∠R + m∠P + m∠Q = 180°. Complete the missing reason in step 5 of the two-column proof.
Statement Reason 1.△RPQ is a triangle. Given 2.Draw line n through Q, parallel to line RP. Construction (Parallel Postulate) 3.∠1 ≅ ∠R and ∠2 ≅ ∠P (∠1 and ∠2 are the angles n makes with QR and QP, on either side of ∠Q). Alternate Interior Angles Theorem 4.m∠1 = m∠R and m∠2 = m∠P. Definition of congruent angles 5.m∠1 + m∠RQP + m∠2 = 180°. 6.m∠R + m∠Q + m∠P = 180°. Substitution Property of Equality 7.m∠R + m∠P + m∠Q = 180°. Commutative Property of Addition - 2.Triangle EFD. Prove: m∠E + m∠F + m∠D = 180°. Complete the missing statement in step 7 of the two-column proof.
Statement Reason 1.△EFD is a triangle. Given 2.Draw line n through D, parallel to line EF. Construction (Parallel Postulate) 3.∠1 ≅ ∠E and ∠2 ≅ ∠F (∠1 and ∠2 are the angles n makes with DE and DF, on either side of ∠D). Alternate Interior Angles Theorem 4.m∠1 = m∠E and m∠2 = m∠F. Definition of congruent angles 5.m∠1 + m∠EDF + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠E + m∠D + m∠F = 180°. Substitution Property of Equality 7. Commutative Property of Addition - 3.Triangle TUS. Prove: m∠T + m∠U + m∠S = 180°. Complete the missing statement in step 4 of the two-column proof.
Statement Reason 1.△TUS is a triangle. Given 2.Draw line n through S, parallel to line TU. Construction (Parallel Postulate) 3.∠1 ≅ ∠T and ∠2 ≅ ∠U (∠1 and ∠2 are the angles n makes with ST and SU, on either side of ∠S). Alternate Interior Angles Theorem 4. Definition of congruent angles 5.m∠1 + m∠TSU + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠T + m∠S + m∠U = 180°. Substitution Property of Equality 7.m∠T + m∠U + m∠S = 180°. Commutative Property of Addition - 4.Triangle FDE. Prove: m∠F + m∠D + m∠E = 180°. Complete the missing statement in step 4 of the two-column proof.
Statement Reason 1.△FDE is a triangle. Given 2.Draw line n through E, parallel to line FD. Construction (Parallel Postulate) 3.∠1 ≅ ∠F and ∠2 ≅ ∠D (∠1 and ∠2 are the angles n makes with EF and ED, on either side of ∠E). Alternate Interior Angles Theorem 4. Definition of congruent angles 5.m∠1 + m∠FED + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠F + m∠E + m∠D = 180°. Substitution Property of Equality 7.m∠F + m∠D + m∠E = 180°. Commutative Property of Addition - 5.Triangle LMK. Prove: m∠L + m∠M + m∠K = 180°. Complete the missing statement in step 1 of the two-column proof.
Statement Reason 1. Given 2.Draw line n through K, parallel to line LM. Construction (Parallel Postulate) 3.∠1 ≅ ∠L and ∠2 ≅ ∠M (∠1 and ∠2 are the angles n makes with KL and KM, on either side of ∠K). Alternate Interior Angles Theorem 4.m∠1 = m∠L and m∠2 = m∠M. Definition of congruent angles 5.m∠1 + m∠LKM + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠L + m∠K + m∠M = 180°. Substitution Property of Equality 7.m∠L + m∠M + m∠K = 180°. Commutative Property of Addition - 6.Triangle DEF. Prove: m∠D + m∠E + m∠F = 180°. Complete the missing statement in step 1 of the two-column proof.
Statement Reason 1. Given 2.Draw line n through F, parallel to line DE. Construction (Parallel Postulate) 3.∠1 ≅ ∠D and ∠2 ≅ ∠E (∠1 and ∠2 are the angles n makes with FD and FE, on either side of ∠F). Alternate Interior Angles Theorem 4.m∠1 = m∠D and m∠2 = m∠E. Definition of congruent angles 5.m∠1 + m∠DFE + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠D + m∠F + m∠E = 180°. Substitution Property of Equality 7.m∠D + m∠E + m∠F = 180°. Commutative Property of Addition - 7.Triangle TUS. Prove: m∠T + m∠U + m∠S = 180°. Complete the missing statement in step 6 of the two-column proof.
Statement Reason 1.△TUS is a triangle. Given 2.Draw line n through S, parallel to line TU. Construction (Parallel Postulate) 3.∠1 ≅ ∠T and ∠2 ≅ ∠U (∠1 and ∠2 are the angles n makes with ST and SU, on either side of ∠S). Alternate Interior Angles Theorem 4.m∠1 = m∠T and m∠2 = m∠U. Definition of congruent angles 5.m∠1 + m∠TSU + m∠2 = 180°. Angles on a straight line sum to 180° 6. Substitution Property of Equality 7.m∠T + m∠U + m∠S = 180°. Commutative Property of Addition - 8.Triangle STU. Prove: m∠S + m∠T + m∠U = 180°. Complete the missing statement in step 4 of the two-column proof.
Statement Reason 1.△STU is a triangle. Given 2.Draw line n through U, parallel to line ST. Construction (Parallel Postulate) 3.∠1 ≅ ∠S and ∠2 ≅ ∠T (∠1 and ∠2 are the angles n makes with US and UT, on either side of ∠U). Alternate Interior Angles Theorem 4. Definition of congruent angles 5.m∠1 + m∠SUT + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠S + m∠U + m∠T = 180°. Substitution Property of Equality 7.m∠S + m∠T + m∠U = 180°. Commutative Property of Addition - 9.Triangle RPQ. Prove: m∠R + m∠P + m∠Q = 180°. Complete the missing statement in step 6 of the two-column proof.
Statement Reason 1.△RPQ is a triangle. Given 2.Draw line n through Q, parallel to line RP. Construction (Parallel Postulate) 3.∠1 ≅ ∠R and ∠2 ≅ ∠P (∠1 and ∠2 are the angles n makes with QR and QP, on either side of ∠Q). Alternate Interior Angles Theorem 4.m∠1 = m∠R and m∠2 = m∠P. Definition of congruent angles 5.m∠1 + m∠RQP + m∠2 = 180°. Angles on a straight line sum to 180° 6. Substitution Property of Equality 7.m∠R + m∠P + m∠Q = 180°. Commutative Property of Addition - 10.Triangle XYZ. Prove: m∠X + m∠Y + m∠Z = 180°. Complete the missing statement in step 1 of the two-column proof.
Statement Reason 1. Given 2.Draw line n through Z, parallel to line XY. Construction (Parallel Postulate) 3.∠1 ≅ ∠X and ∠2 ≅ ∠Y (∠1 and ∠2 are the angles n makes with ZX and ZY, on either side of ∠Z). Alternate Interior Angles Theorem 4.m∠1 = m∠X and m∠2 = m∠Y. Definition of congruent angles 5.m∠1 + m∠XZY + m∠2 = 180°. Angles on a straight line sum to 180° 6.m∠X + m∠Z + m∠Y = 180°. Substitution Property of Equality 7.m∠X + m∠Y + m∠Z = 180°. Commutative Property of Addition
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