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Parallel lines and Transversals
Concept 18 Parallel lines and Transversals
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Corresponding Angles Postulate
If _______ parallel lines are cut by a ________________ then each pair of ________________ angles are ___________. two transversal corresponding congruent
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Given: πβ 11=34Β° Prove: πβ 15=34Β° 1. 2. 3. 4. πβ 11=34Β° Given
Statements Reasons 1. 2. 3. 4. πβ 11=34Β° Given β 11β
β 15 Cooresponding β Post. Def. of Congruent β πβ 11=πβ 15 πβ 15=34Β° Substitution Prop.
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1. 2. 3. 4. 5. 6. Given: π || π, πβ 11=51Β° Prove: πβ 16=51Β° Statements
Reasons 1. 2. 3. 4. 5. 6. 7. π || π Given πβ 11=51Β° Given Cooresponding β Post. β 11β
β 15 β 15β
β 16 Vertical Angles Thm. Transitive Prop. β 11β
β 16 πβ 11=πβ 16 Def. of Cong. Segments πβ 16=51Β° Substitution Prop.
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Given: l || m Prove: β 3 β β 7 Alternate Interior Angles Theorem
Statements Reasons 1. 2. 3. 4. π || π Given β 3β
β 5 Cooresponding β Post. β 5β
β 7 Vertical Angles Thm. β 3β
β 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
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Given: j || k, mβ 1=126Β°, πβ 7=7(π₯ β7) Prove: x = 25
Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. j || π Given β 7β
β 1 Cooresponding β Post. πβ 7=πβ 1 Def. of Congruent Angles πβ 1=126, πβ 7=7(π₯β7) Givens 7(π₯β7)=126 Substitution Prop. 7π₯β49=126 Distributive Prop. 7π₯=175 Addition Prop. π₯=25 Division Prop
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Alternate Exterior Angles Theorem
Given: j || k Prove: β 1 β
β 2 Statements Reasons 1. 2. 3. 4. j || π Given β 1β
β 3 Cooresponding β Post. β 3β
β 2 Vertical Angles Thm. Transitive Prop. β 1β
β 2 Alternate Exterior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ___________________ angles are ____________. two transversal alternate exterior congruent
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1. 2. 3. 4. 5. Given: l || m, p || q Prove: β 1 β β 3 Statements Reasons
β 1β
β 2 Alt. Ext. Angles Thm. π || π Given Corresponding Angles Post. β 2β
β 3 β 1β
β 3 Transitive Prop.
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Prove: β 1 and β 2 are supplementary Statements Reasons 1. 2. 3. 4. 5.
Given: j || k Prove: β 1 and β 2 are supplementary Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. j || π Given β 1β
β 3 Cooresponding β Post. πβ 1=πβ 3 Def. of Congruent Angles β 2 πππ β 3 form a linear pair Def. of Linear Pair/Given β 2 πππ β 3 are supplementary Linear Pair Thm πβ 2+πβ 3=180 Def. of Supplementary πβ 2+πβ 1=180 Substitution Prop. β 1 πππ β 2 are supplementary Def of Supplementary.
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Same Side Interior Angle Theorem
If _______ parallel lines are cut by a ______________ then each pair of ___________________________ angles are __________________. two transversal same side interior supplementary
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Given: p || q Prove: x = 7 Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. π || π Given β 2 πππ β 3 are supplementary Same Side Int. Angles Thm. πβ 2+πβ 3=180 Def. of Supplementary Angles β 1β
β 3 Vertical Angles Thm. πβ 1=πβ 3 Def. of Congruent Angles πβ 2+πβ 1=180 Substitution Property πβ 1=94, πβ 2=13π₯β5 Givens Substitution Property 13π₯β5+94=180 13π₯+89=180 Simplify 13π₯=91 Subtraction Prop. π₯=7 Division Prop
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1. 2. 3. 4. 5. 6. 7. 8. Given: j || k Prove: β 1 & β 3 are supplementary
Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. j || π Given β 1β
β 2 Corresponding Angles Post. πβ 1=π β 2 Def. of Congruent Angles β 2 πππ β 3 are a linear pair Def of Linear Pair/Given β 2 πππ β 3 are supplementary Linear Pair Post. πβ 2+πβ 3=180 Definition of Supp. Angles πβ 1+πβ 3=180 Substitution Prop. β 1 πππ β 3 are supplementary Def. of Supplementary
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Same Side Exterior Angles Theorem
If _______ parallel lines are cut by a ________________ then each pair of ___________________________ angles are __________________. two transversal same side exterior supplementary
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Given: p || q Prove: x = 23 Statements Reasons 1. 2. 3. 4. 5. π || π
5π₯β24+89=180 Same Side Ext. Angle Thm 5π₯+65=180 Simplify 5π₯=115 Subtraction Property π₯=23 Division Prop
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Given: p || q and p ο r Prove: q ο r 1. 2. 3. 4. 5. 6. 7. 8.
Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9. π || π Given β 1β
β 2 Corresponding Angles Post. π β₯π Given β 2 is a right angle Def of perpendicular πβ 2=90 Def of right angle πβ 1=πβ 2 Def. of Congruent Angles πβ 1=90 Substitution Prop. β 1 is a right angle Def. of Right angle π β₯π Def. of perpendicular
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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.
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Prove: β C is a right angle 1. 2. 3. 4.
Given: π΄π΅ || π·πΆ , π΄π΅ β₯ π΅πΆ Prove: β C is a right angle Statements Reasons 1. 2. 3. 4. π΄π΅ || π·πΆ Given π΄π΅ β₯ π΅πΆ Given π·πΆ β₯ π΅πΆ Perp. Transversal Thm. Def. of perpendicular β C is a right angle
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Find the measure of each angle.
Given: ππ || π
π , πΏπ β₯ ππΎ Find the measure of each angle. πβ 1= 35Β° πβ 2= 180 β35 =145Β° πβ 3= 35Β° 125Β° πβ 4= 90 β55 =35Β° πβ 5= 55Β° πβ 6= 180 β125 =55Β° πβ 7= 125Β°
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Given: π΄π΅ || πΆπ· , π΄πΆ || π·πΉ , πΆπ· || πΈπΉ Prove: β π΅π΄πΆβ
β πΈπΉπ·
Statements Reasons 1. 2. 3. 4. 5. 6. 7. 8. π΄π΅ || πΆπ· Given β π΅π΄πΆβ
β π·πΆπ΄ Alt. Int. Angles Thm. π΄πΆ || π·πΉ Given β π·πΆπ΄β
β CDF Alt. Int. Angles Thm. β π΅π΄πΆβ
β CDF Transitive Prop. πΆπ· || πΈπΉ Given β πΆπ·πΉβ
β EFD Alt. Int. Angles Thm. β π΅π΄πΆβ
β EFD Transitive Prop.
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In the figure, mβ 9 = 80 and mβ 5 = 68. Find the measure of each angle.
1. β 12 = β 1 = 3. β 4 = β 3 = 5. β 7 = β 16 = 180 β 80 80 = 100 80 100 80 68 68 180 β 68 = 112
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7. In the figure, mο11 = 51. Find mο15.
9. If mο2 = 125, find mο3. 10. Find mο4. πβ ππ=ππΒ° πβ π=πππΒ° 8. Find mο16. πβ ππ=πππβπππΒ° πβ ππ=ππΒ° =ππΒ°
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11. If mο5 = 2x β 10, and mο7 = x + 15, find x.
12. If mο4 = 4(y β 25), and mο8 = 4y, find y. πβ π=πβ π ππβππ=π+ππ πβππ=ππ π=ππ πβ π+πβ π=πππ 4 πβππ +ππ=πππ ππβπππ+ππ=πππ ππβπππ=πππ ππ=πππ π=ππ
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If mο1 = 9x + 6 and mο2 = 2(5x β 3) find x.
14. mο3 = 5y + 14 to find y. πβ π=πβ π ππ+π=π(ππβπ) ππ+π=πππβπ π=πβπ 12=π πβ π=πβ π π ππβπ =ππ+ππ π ππβπ =πππβπ πππ=ππ+ππ =ππ(ππ)βπ πππ=ππ =πππβπ ππ=π =πππ
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Find the value of the variable(s) in each figure
Find the value of the variable(s) in each figure. Explain your reasoning. 106 =ππ ππ+ππ+π=πππ ππ+ππ=πππ π+πππ=πππ ππ+π=πππ ππ=ππ π=ππ ππ=πππ π=ππ π=ππ ππ+πππ=πππ ππ+π=πππ ππ=ππ ππ=ππ π=ππ π=πππ π=ππ
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Proving Lines are parallel
Concept 19
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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.
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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.
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Given: β 1 β β 2 Prove: l || m β 3β
β 1 Vertical Angles Thm Given β 1β
β 2 β 3β
β 2 Transitive Prop. π || π Corresponding Angles Converse Postulate
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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.
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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 = 190 and Same Side Interior Angles Converse Thm says they should add to 180.
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Given: mβ 1= 135, mβ 4 = 45 Prove: n || o 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.
6. 7. πβ 1=135 Givens πβ 4=45 Addition Prop. πβ 1+ β 4=180 Vertical Angles Thm β 1β
β 2 πβ 1=π β 2 Def. of Congruent Angles Substitution Prop. πβ 2+ β 4=180 β 2 & β 4 are supp. Def of supp. π || π Same Side Interior Angles Converse Theorem
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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.
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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.
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Given: β 1 β β 2 Prove: l || m β 3β
β 1 Vertical Angles Thm Given β 1β
β 2 β 3β
β 2 Transitive Prop. l || m Alternate Interior Angles Converse Thm.
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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.
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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.
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Given: β 3 β β 2 Prove: l || m Vertical Angles Thm β 1β
β 3 β 3β
β 2 Given β 1β
β 2 Transitive Prop. π || π Alternate Exterior Angles Converse Thm.
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