geometry unit 3 quiz 2

Converse of the Alternate Exterior Angles Theorem

If two coplanar lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

Converse of the Alternate Interior Angles Theorem

If two coplanar lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

Converse of the Corresponding Angles Postulate

If two coplanar lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

Converse of the Same-Side Exterior Angles Theorem

If two coplanar lines are cut by a transversal so that same-side exterior angles are supplementary, then the lines are parallel.

Converse of the Same-Side Interior Angles Theorem

If two coplanar lines are cut by a transversal so that same-side interior angles are supplementary, then the lines are parallel.

Theorem 3.4C

If two coplanar lines are perpendicular to the same line, then the two lines are parallel

Theorem 3.4A

If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.

Perpendicular Transversal Theorem

In a plane, if two parallel lines are cut by a transversal such that the transversal is perpendicular to one of the lines, then it is perpendicular to the other line.

Given: lines ℓ and m cut by transversal, t alternate interior angles ∠4≅∠5 Prove: ℓ∥m

1. lines ℓ and m cut by transversal, t alternate interior angles ∠4≅∠5 1. Given 2. ∠1≅∠4 2. Vert. ∠s Thrm. 3. ∠1≅∠5 3. Trans. Property of ≅ 4. ℓ∥m 4. Conv. Corr. ∠s Post.

Given: lines ℓ and m cut by transversal, t; ∠2 is supplementary to ∠8 . Prove: ℓ ∥ m

1. lines ℓ and m cut by transversal, t; ∠2 is supplementary to ∠8 1. Given 2. m∠2 + m∠8=180∘ 2. Def. supplementary angles 3. ∠6 and ∠8 are a linear pair of angles 3. Def. linear pair of angles 4. ∠6 and ∠8 are supplementary 4.Linear Pair Theorem 5. m∠6 + m∠8= 180∘ 5. Def. supplementary angles 6. m∠2 + m∠8=m∠6 + m∠8 6. Substitution 7. m∠2=m∠6 7. Subtraction Prop. of Equality 8. ∠2≅∠6 8. Def. congruent angles 9. ℓ ∥ m 9. Converse of the Corresponding Angles Postulate

Given: lines ℓ and m cut by transversal, t∠2 is supplementary to ∠5. Prove: ℓ∥m

1. lines ℓ and m cut by transversal, t∠2 is supplementary to ∠5. 1. Given 2. m∠2 + m∠5=180° 2. Def. of supp. ∠s 3. ∠5 are ∠6 linear pairs of angles. 3. Def. of linear pairs of ∠s 4. ∠5 is supplementary to ∠6. 4. Linear Pair Theorem 5. m∠5 + m∠6=180° 5. Def. of supp. ∠s 6. m∠2+m∠5=m∠5+m∠6 6. Trans. Prop. of = 7. m∠2=m∠6 7. Subtraction Prop. of = 8. ∠2≅∠6 8. Def. of ≅∠s 9. ℓ∥m 9. Conv. Corr. ∠s Post.

Given: m intersects ℓ; ∠1≅∠2. Prove: ℓ⊥m

1. m intersects ℓ; ∠1≅∠2. 1. Given 2. m∠1=m∠2 2. Def. of ≅∠s 3. ∠1 and ∠2 are a linear pair of angles. 3. Def. of linear pairs of ∠s 4. ∠1 is supplementary to ∠2. 4. Linear Pair Thrm. 5. m∠1 + m∠2=180° 5. Def. of supp. ∠s 6. m∠1 + m∠1=180° 6. Substitution Prop. of = 7. 2(m∠1)=180° 7. Simplify 8. m∠1=90° 8. Divisions Prop. of = 9. m∠2=90° 9. Substitution Prop. of = 10. ℓ⊥m 10. Def. ⊥ lines

Given: t⊥ℓ; t⊥m Prove: ℓ∥m

1. t⊥ℓ; t⊥m 1. Given 2. ∠1 and ∠2 are rt. ∠s. 2. Def. of ⊥ lines 3. ∠1 and ∠2 are corresponding ∠s. 3. Def. of Corr. ∠s 4. ∠1≅∠2 4. Rt. ∠≅ Thrm. 5. ℓ∥m 5. Conv. of Corr. ∠s Post.

Given: ℓ∥m; t⊥ℓ Prove: t⊥m

1. ℓ∥m; t⊥ℓ 1. Given 2. ∠2≅∠6 2. Corr. ∠s Post. 3. m∠2=m∠6 3. Def. of ≅∠s 4. m∠2=90° 4. Def. ⊥ lines 5. m∠6=90° 5. Trans. Prop. of = 6. t⊥m 6. Def. ⊥ lines

Given: lines ℓ and m cut by transversal, t, alternate exterior angles ∠1≅∠8 Prove: ℓ ∥ m

1.lines ℓ and m cut by transversal, t, alternate exterior angles ∠1≅∠8 1. Given 2. ∠5≅∠8 2. Vertical Angles Theorem 3. ∠1≅∠5 3. Transitive Property of Congruence 4. ℓ ∥ m 4. Converse of the Corresponding Angles Postulate