Parallel lines and Transversals Concept 18 Parallel lines and Transversals

Corresponding Angles Postulate If _______ parallel lines are cut by a ________________ then each pair of ________________ angles are ___________. two transversal corresponding congruent

Given: 𝑚∠11=34° Prove: 𝑚∠15=34° 1. 2. 3. 4. 𝑚∠11=34° Given Statements Reasons 1. 2. 3. 4. 𝑚∠11=34° Given ∠11≅∠15 Cooresponding ∠ Post. Def. of Congruent ∠ 𝑚∠11=𝑚∠15 𝑚∠15=34° Substitution Prop.

1. 2. 3. 4. 5. 6. Given: 𝑝 || 𝑞, 𝑚∠11=51° Prove: 𝑚∠16=51° Statements Reasons 1. 2. 3. 4. 5. 6. 7. 𝑝 || 𝑞 Given 𝑚∠11=51° Given Cooresponding ∠ Post. ∠11≅∠15 ∠15≅∠16 Vertical Angles Thm. Transitive Prop. ∠11≅∠16 𝑚∠11=𝑚∠16 Def. of Cong. Segments 𝑚∠16=51° Substitution Prop.

Given: l || m Prove: ∠3 ≌ ∠7 Alternate Interior Angles Theorem Statements Reasons 1. 2. 3. 4. 𝑙 || 𝑚 Given ∠3≅∠5 Cooresponding ∠ Post. ∠5≅∠7 Vertical Angles Thm. ∠3≅∠7 Transitive Prop. Alternate Interior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ______________________ angles are _____________. two transversal alternate interior congruent

Given: j || k, m∠1=126°, 𝑚∠7=7(𝑥 −7) Prove: x = 25 Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. j || 𝑘 Given ∠7≅∠1 Cooresponding ∠ Post. 𝑚∠7=𝑚∠1 Def. of Congruent Angles 𝑚∠1=126, 𝑚∠7=7(𝑥−7) Givens 7(𝑥−7)=126 Substitution Prop. 7𝑥−49=126 Distributive Prop. 7𝑥=175 Addition Prop. 𝑥=25 Division Prop

Alternate Exterior Angles Theorem Given: j || k Prove: ∠1 ≅∠2 Statements Reasons 1. 2. 3. 4. j || 𝑘 Given ∠1≅∠3 Cooresponding ∠ Post. ∠3≅∠2 Vertical Angles Thm. Transitive Prop. ∠1≅∠2 Alternate Exterior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ___________________ angles are ____________. two transversal alternate exterior congruent

1. 2. 3. 4. 5. Given: l || m, p || q Prove: ∠1 ≌ ∠3 Statements Reasons ∠1≅∠2 Alt. Ext. Angles Thm. 𝑝 || 𝑞 Given Corresponding Angles Post. ∠2≅∠3 ∠1≅∠3 Transitive Prop.

Prove: ∠1 and ∠2 are supplementary Statements Reasons 1. 2. 3. 4. 5. Given: j || k Prove: ∠1 and ∠2 are supplementary Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. j || 𝑘 Given ∠1≅∠3 Cooresponding ∠ Post. 𝑚∠1=𝑚∠3 Def. of Congruent Angles ∠2 𝑎𝑛𝑑 ∠3 form a linear pair Def. of Linear Pair/Given ∠2 𝑎𝑛𝑑 ∠3 are supplementary Linear Pair Thm 𝑚∠2+𝑚∠3=180 Def. of Supplementary 𝑚∠2+𝑚∠1=180 Substitution Prop. ∠1 𝑎𝑛𝑑 ∠2 are supplementary Def of Supplementary.

Same Side Interior Angle Theorem If _______ parallel lines are cut by a ______________ then each pair of ___________________________ angles are __________________. two transversal same side interior supplementary

Given: p || q Prove: x = 7 Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 𝑝 || 𝑞 Given ∠2 𝑎𝑛𝑑 ∠3 are supplementary Same Side Int. Angles Thm. 𝑚∠2+𝑚∠3=180 Def. of Supplementary Angles ∠1≅∠3 Vertical Angles Thm. 𝑚∠1=𝑚∠3 Def. of Congruent Angles 𝑚∠2+𝑚∠1=180 Substitution Property 𝑚∠1=94, 𝑚∠2=13𝑥−5 Givens Substitution Property 13𝑥−5+94=180 13𝑥+89=180 Simplify 13𝑥=91 Subtraction Prop. 𝑥=7 Division Prop

1. 2. 3. 4. 5. 6. 7. 8. Given: j || k Prove: ∠1 & ∠3 are supplementary Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. j || 𝑘 Given ∠1≅ ∠2 Corresponding Angles Post. 𝑚∠1=𝑚 ∠2 Def. of Congruent Angles ∠2 𝑎𝑛𝑑 ∠3 are a linear pair Def of Linear Pair/Given ∠2 𝑎𝑛𝑑 ∠3 are supplementary Linear Pair Post. 𝑚∠2+𝑚∠3=180 Definition of Supp. Angles 𝑚∠1+𝑚∠3=180 Substitution Prop. ∠1 𝑎𝑛𝑑 ∠3 are supplementary Def. of Supplementary

Same Side Exterior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ___________________________ angles are __________________. two transversal same side exterior supplementary

Given: p || q Prove: x = 23 Statements Reasons 1. 2. 3. 4. 5. 𝑝 || 𝑞 5𝑥−24+89=180 Same Side Ext. Angle Thm 5𝑥+65=180 Simplify 5𝑥=115 Subtraction Property 𝑥=23 Division Prop

Given: p || q and p  r Prove: q  r 1. 2. 3. 4. 5. 6. 7. 8. Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9. 𝑝 || 𝑞 Given ∠1≅ ∠2 Corresponding Angles Post. 𝑝 ⊥𝑟 Given ∠2 is a right angle Def of perpendicular 𝑚∠2=90 Def of right angle 𝑚∠1=𝑚∠2 Def. of Congruent Angles 𝑚∠1=90 Substitution Prop. ∠1 is a right angle Def. of Right angle 𝑞 ⊥𝑟 Def. of perpendicular

Perpendicular Transversal Theorem If two parallel lines are cut by a transversal and one line is perpendicular to the transversal, then the other line is perpendicular to the transversal.

Prove: ∠C is a right angle 1. 2. 3. 4. Given: 𝐴𝐵 || 𝐷𝐶 , 𝐴𝐵 ⊥ 𝐵𝐶 Prove: ∠C is a right angle Statements Reasons 1. 2. 3. 4. 𝐴𝐵 || 𝐷𝐶 Given 𝐴𝐵 ⊥ 𝐵𝐶 Given 𝐷𝐶 ⊥ 𝐵𝐶 Perp. Transversal Thm. Def. of perpendicular ∠C is a right angle

Find the measure of each angle. Given: 𝑃𝑄 || 𝑅𝑆 , 𝐿𝑀 ⊥ 𝑁𝐾 Find the measure of each angle. 𝑚∠1= 35° 𝑚∠2= 180 −35 =145° 𝑚∠3= 35° 125° 𝑚∠4= 90 −55 =35° 𝑚∠5= 55° 𝑚∠6= 180 −125 =55° 𝑚∠7= 125°

Given: 𝐴𝐵 || 𝐶𝐷 , 𝐴𝐶 || 𝐷𝐹 , 𝐶𝐷 || 𝐸𝐹 Prove: ∠𝐵𝐴𝐶≅∠𝐸𝐹𝐷 Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. 𝐴𝐵 || 𝐶𝐷 Given ∠𝐵𝐴𝐶≅ ∠𝐷𝐶𝐴 Alt. Int. Angles Thm. 𝐴𝐶 || 𝐷𝐹 Given ∠𝐷𝐶𝐴≅ ∠CDF Alt. Int. Angles Thm. ∠𝐵𝐴𝐶≅ ∠CDF Transitive Prop. 𝐶𝐷 || 𝐸𝐹 Given ∠𝐶𝐷𝐹≅ ∠EFD Alt. Int. Angles Thm. ∠𝐵𝐴𝐶≅ ∠EFD Transitive Prop.

In the figure, m∠9 = 80 and m∠5 = 68. Find the measure of each angle. 1. ∠12 = 2. ∠1 = 3. ∠4 = 4. ∠3 = 5. ∠7 = 6. ∠16 = 180 – 80 80 = 100 80 100 80 68 68 180 – 68 = 112

7. In the figure, m11 = 51. Find m15. 9. If m2 = 125, find m3. 10. Find m4. 𝒎∠𝟏𝟓=𝟓𝟏° 𝒎∠𝟑=𝟏𝟐𝟓° 8. Find m16. 𝒎∠𝟏𝟒=𝟏𝟖𝟎−𝟏𝟐𝟓° 𝒎∠𝟏𝟔=𝟓𝟏° =𝟓𝟓°

11. If m5 = 2x – 10, and m7 = x + 15, find x. 12. If m4 = 4(y – 25), and m8 = 4y, find y. 𝒎∠𝟓=𝒎∠𝟕 𝟐𝒙−𝟏𝟎=𝒙+𝟏𝟓 𝒙−𝟏𝟎=𝟏𝟓 𝒙=𝟐𝟓 𝒎∠𝟒+𝒎∠𝟖=𝟏𝟖𝟎 4 𝒚−𝟐𝟓 +𝟒𝒚=𝟏𝟖𝟎 𝟒𝒚−𝟏𝟎𝟎+𝟒𝒚=𝟏𝟖𝟎 𝟖𝒚−𝟏𝟎𝟎=𝟏𝟖𝟎 𝟖𝒚=𝟐𝟖𝟎 𝒚=𝟑𝟓

If m1 = 9x + 6 and m2 = 2(5x – 3) find x. 14. m3 = 5y + 14 to find y. 𝒎∠𝟏=𝒎∠𝟐 𝟗𝒙+𝟔=𝟐(𝟓𝒙−𝟑) 𝟗𝒙+𝟔=𝟏𝟎𝒙−𝟔 𝟔=𝒙−𝟔 12=𝒙 𝒎∠𝟐=𝒎∠𝟑 𝟐 𝟓𝒙−𝟑 =𝟓𝒚+𝟏𝟒 𝟐 𝟓𝒙−𝟑 =𝟏𝟎𝒙−𝟔 𝟏𝟏𝟒=𝟓𝒚+𝟏𝟒 =𝟏𝟎(𝟏𝟐)−𝟔 𝟏𝟎𝟎=𝟓𝒚 =𝟏𝟐𝟎−𝟔 𝟐𝟎=𝒚 =𝟏𝟏𝟒

Find the value of the variable(s) in each figure Find the value of the variable(s) in each figure. Explain your reasoning. 15. 16. 106 =𝟑𝟎 𝟐𝒙+𝟗𝟎+𝒙=𝟏𝟖𝟎 𝟑𝒙+𝟗𝟎=𝟏𝟖𝟎 𝒙+𝟏𝟎𝟔=𝟏𝟖𝟎 𝟒𝒛+𝟔=𝟏𝟎𝟔 𝟑𝒙=𝟗𝟎 𝒙=𝟕𝟒 𝟒𝒛=𝟏𝟎𝟎 𝒙=𝟑𝟎 𝒛=𝟐𝟓 𝟐𝒚+𝟏𝟎𝟔=𝟏𝟖𝟎 𝟑𝟎+𝒛=𝟏𝟖𝟎 𝟐𝒚=𝟑𝟎 𝟐𝒚=𝟕𝟒 𝒚=𝟑𝟕 𝒛=𝟏𝟓𝟎 𝒚=𝟏𝟓

Proving Lines are parallel Concept 19

Corresponding Angles Converse Postulate Same Side Interior Angles Converse Theorem Alternate Interior Angles Converse Theorem Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel.

Is it possible to prove that line p and q are parallel Is it possible to prove that line p and q are parallel? If so explain how. Yes, because 70 =70 and the Corresponding Angles Converse Post.

Given: ∠1 ≌ ∠2 Prove: l || m 1. 1. 2. 2. 3. 3. 4. 4. ∠3≅ ∠1 Vertical Angles Thm Given ∠1≅ ∠2 ∠3≅ ∠2 Transitive Prop. 𝑙 || 𝑚 Corresponding Angles Converse Postulate

Corresponding Angles Converse Postulate Same Side Interior Angles Converse Theorem Alternate Interior Angles Converse Theorem Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal and same side interior angles (angles that lie between the two lines and are on the same side of the transversal) are supplementary, then the lines are parallel If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel.

Is it possible to prove that line p and q are parallel Is it possible to prove that line p and q are parallel? If so explain how. NO, because using vertical angles the 75 would then make a same side interior angle pair with the 115. 115+75 = 190 and Same Side Interior Angles Converse Thm says they should add to 180.

Given: m∠1= 135, m∠4 = 45 Prove: n || o 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.   1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 7. 𝑚∠1=135 Givens 𝑚∠4=45 Addition Prop. 𝑚∠1+ ∠4=180 Vertical Angles Thm ∠1≅ ∠2 𝑚∠1=𝑚 ∠2 Def. of Congruent Angles Substitution Prop. 𝑚∠2+ ∠4=180 ∠2 & ∠4 are supp. Def of supp. 𝑝 || 𝑞 Same Side Interior Angles Converse Theorem

Corresponding Angles Converse Postulate Same Side Interior Angles Converse Theorem Alternate Interior Angles Converse Theorem Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal and same side interior angles (angles that lie between the two lines and are on the same side of the transversal) are supplementary, then the lines are parallel If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate interior angles (angles that lie between the two lines and on opposite sides of the transversal) are congruent, then the lines are parallel.

Is it possible to prove that line p and q are parallel Is it possible to prove that line p and q are parallel? If so explain how. Yes, because 115 =115 and the Alternate Interior Angles Converse Thm.

Given: ∠1 ≌ ∠2 Prove: l || m 1. 1. 2. 2. 3. 3. 4. 4. ∠3≅ ∠1 Vertical Angles Thm Given ∠1≅ ∠2 ∠3≅ ∠2 Transitive Prop. l || m Alternate Interior Angles Converse Thm.

Corresponding Angles Converse Postulate Same Side Interior Angles Converse Theorem Alternate Interior Angles Converse Theorem Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal and same side interior angles (angles that lie between the two lines and are on the same side of the transversal) are supplementary, then the lines are parallel If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate interior angles (angles that lie between the two lines and on opposite sides of the transversal) are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate exterior angles (angles that lie outside the two lines and on opposite sides of the transversal) are congruent, then the lines are parallel.

Is it possible to prove that line p and q are parallel Is it possible to prove that line p and q are parallel? If so explain how. Yes, because 75 =75 and the Alternate Exterior Angles Converse Thm.

Given: ∠3 ≌ ∠2 Prove: l || m 1. 1. 2. 2. 3. 3. 4. 4. Vertical Angles Thm ∠1≅ ∠3 ∠3≅ ∠2 Given ∠1≅ ∠2 Transitive Prop. 𝑙 || 𝑚 Alternate Exterior Angles Converse Thm.