Geometry Student Text 2nd Edition Chapter 2 Parallel and Perpendicular Lines
Carnegie Learning Geometry Student Text 2nd Edition Chapter 2 Exercise 2.4 Solution Page 98 Problem 1 Answer
The statement given is that if two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.
The hypothesis (p) of the statement is that’ If two parallel lines are intersected by a transversal’.
The hypothesis (p) of the statement is that’ If two parallel lines are intersected by a transversal’.
Page 98 Problem 2 Answer
The statement given is that if two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.
The conclusion (q) of the statement is that’ then the alternate interior angles are congruent’.
The conclusion (q) of the given statement is : ‘then the alternate interior angles are congruent’.
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Page 98 Problem 3 Answer
The statement given is that if two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.
The Alternate Interior Angle Converse Conjecture of the statement is that ‘If two lines intersected by a transversal form alternate interior angles, then the lines are parallel.’
The Alternate Interior Angle Converse Conjecture of the given statement is that ‘If two lines intersected by a transversal form alternate interior angles, then the lines are parallel.’
Page 98 Problem 4 Answer
The statement given is that if two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent.
The hypothesis (p) of the statement is that’ If two parallel lines are intersected by a transversal’.
Hence, the hypothesis (p) of the statement is that’ If two parallel lines are intersected by a transversal’.
Page 98 Problem 5 Answer
The statement given is that if two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent.
The conclusion (q) of the statement is that ‘ the alternate exterior angles are congruent ‘.
Hence, the conclusion (q) of the statement is that ‘ the alternate exterior angles are congruent ‘.
Solutions For Parallel And Perpendicular Lines Exercise 2.4 In Carnegie Learning Geometry Page 98 Problem 6 Answer
The statement is given:
Page 98 Problem 7 Answer
The statement is given:
Same-Side Interior Angle Theorem: if two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary.
The hypothesis (p) of the statement is that ‘ if two parallel lines are intersected by a transversal.’
Hence, the hypothesis (p) of the statement is that ‘ if two parallel lines are intersected by a transversal.’
Page 98 Problem 8 Answer
The statement given :
Same-Side Interior Angle Theorem: if two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary.
The conclusion (q) of the statement is that ‘ the same-side interior angles are supplementary.’
Hence, the conclusion (q) of the statement is that ‘ the same-side interior angles are supplementary.’
Page 98 Problem 9 Answer
For a statement: if p then q the converse of the statement is if q then p
p is the hypothesis and q is the conclusion
Now we analyze the given statement
Same-Side Interior Angle Theorem: If two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary.
The hypothesis p is ‘two parallel lines are intersected by a transversal ‘
Conclusion (q) is ‘same side interior angles are supplementary ‘
The converse is If a transversal intersects two lines such that same side interior angles are supplementary then the two lines are parallel
The hypothesis p is ‘two parallel lines are intersected by a transversal ‘
Conclusion (q) is ‘same side interior angles are supplementary ‘
Same-Side Interior Angle Converse Conjecture: If a transversal intersects two lines such that same-side interior angles are supplementary then the two lines are parallel
Carnegie Learning Geometry 2nd Edition Exercise 2.4 Solutions Page 98 Problem 10 Answer
Given: Same-Side Exterior Angle Theorem: If two parallel lines are intersected by a transversal, then the same-side exterior angles are supplementary.
For a statement : if p then q the converse of the statement is if q then p
p is the hypothesis and q is the conclusion
Now we analyze the given statement
If two parallel lines are intersected by a transversal, then the same-side exterior angles are supplementary.
The hypothesis p is ‘two parallel lines are intersected by a transversal ‘
Hypothesis p: ‘two parallel lines are intersected by a transversal ‘
Page 98 Problem 11 Answer
For a statement: if p then q the converse of the statement is if q then p
p is the hypothesis and q is the conclusion
Now we analyze the given statement
Same-Side Interior Angle Theorem: If two parallel lines are intersected by a transversal, then the same-side exterior angles are supplementary.
Conclusion (q) is ‘same side exterior angles are supplementary ‘
Conclusion q: ‘same side exterior angles are supplementary ‘
Page 98 Problem 12 Answer
Given: Same-Side Exterior Angle Theorem: If two parallel lines are intersected by a transversal, then the same-side exterior angles are supplementary.
For a statement: if p then q the converse of the statement is if q then p
p is the hypothesis and q is the conclusion
Now we analyze the given statement
If two parallel lines are intersected by a transversal, then the same-side exterior angles are supplementary.
The hypothesis p is ‘two parallel lines are intersected by a transversal ‘
Conclusion (q) is ‘the same-side exterior angles are supplementary ‘
The converse is: If a transversal intersects two lines such that same side exterior angles are supplementary then the two lines are parallel
Same-Side Exterior Angle Converse Conjecture:
If a transversal intersects two lines such that same side exterior angles are supplementary then the two lines are parallel
Parallel And Perpendicular Lines Solutions Chapter 2 Exercise 2.4 Carnegie Learning Geometry Page 99 Problem 13 Answer
Given: line r and transversal s the Corresponding Angle Converse Postulate says If two lines intersected by a transversal
We need two lines that is intersected by a transversal. Here we have only line ‘r’ that is intersected by transversal ‘s’
(1)Construct another line t
(2) The line t forms congruent corresponding angles
(3) We can say that the lines r and t are parallel
Steps are :
(1)Construct a another line t
(2) The line t forms congruent corresponding angles
(3) We can say that the lines r and t are parallel
Page 99 Problem 14 Answer
Given: line r and transversal s from first part of the question
the Corresponding Angle Converse Postulate says If two lines intersected by a transversal
We need two lines that is intersected by a transversal. Here we have only line r that is intersected by transversal s
Construct a another line t . The line t forms congruent corresponding angles. We can say that the lines r and t are parallel
From the we can say that , line s is the transversal line s is the transversal line
Page 99 Problem 15 Answer
Given : line r and transversal s from first part of the question
the Corresponding Angle Converse Postulate says If two lines intersected by a transversal
We need two lines that is intersected by a transversal.
Here we have only line r that is intersected by transversal s
Construct a another line t . The line t forms congruent corresponding angles . We can say that the lines r and t are parallel
From the we can say that , line s is the transversal
Parallel lines are line r and t
Parallel lines are line r and t
Step-By-Step Solutions For Carnegie Learning Geometry Chapter 2 Exercise 2.4 Page 101 Problem 16 Answer
The Alternate exterior Angle Converse Conjecture states: “If two lines intersected by a transversal form congruent alternate exterior angles, then the lines are parallel.”
To prove this, we need to consider that two lines w and x form congruent alternate exterior angles
Given: ∠2≅∠7 ( alternate exterior angles 2 and 7 are congruent )
Prove: w∥x ( line w is parallel to line x)
Given: ∠2≅∠7 alternate exterior angles are congruent
Prove: w∥x lines w and x are parallel.
Page 101 Problem 17 Answer
If two lines intersected by a transversal form congruent alternate exterior angles, then the lines are parallel.
Given that congruent alternate exterior angles∠1=∠8
∠2=∠7
We have to show that the Alternate Exterior Angle Converse Conjecture.
We have
Also, alternate exterior angles are congruent. So, ∠1=∠8
∠2=∠7
But angles∠5 and ∠8 are vertically opposite angles, that is∠5=∠8 and also∠1=∠4 Vertically opposite angles are equal.
We get∠4=∠5 these are alternate interior angles.
When alternate angles are equal, then the lines are parallel.
Hence, the statement “If two lines intersected by a transversal form congruent alternate exterior angles, then the lines are parallel.” is proved.
Carnegie Learning Geometry Chapter 2 Exercise 2.4 Free Solutions Page 102 Problem 18 Answer
Given that If two lines intersect by a transversal form supplementary same-side interior angles, then the lines are parallel.
Supplementary same-side interior angles that is∠4+∠6=180∘ and ∠3+∠5=180∘
We have to prove that statements for the Same Side Interior Angle Converse Conjecture.
We have
Also, given that∠4+∠6=180∘ −−−−−−−−(1) and ∠3+∠5=180∘
If a ray stands on a line, then the angles so formed is linear pair are parallel, then∠2+∠4=180∘−−−−−−−−(2)
From(1) and (2)
We get,∠4+∠6=∠2+∠4
∠6=∠2
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal. Then, the lines are parallel.
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal. Then, the lines are parallel.
Hence, the statement “If two lines intersected by a transversal form supplementary same-side interior angles, then the lines are parallel.” is proved.
Page 102 Problem 19 Answer
Given that If two lines intersected by a transversal form supplementary same-side interior angles, then the lines are parallel.
Also∠4+∠6=180∘ and ∠3+∠5=180∘
We have to prove the Same-Side Interior Angle Converse Conjecture.
We have
Also, given that∠4+∠6=180∘ −−−−−−−−(1) and ∠3+∠5=180∘
If a ray stands on a line, then the angles so formed is linear pair ∠2+∠4=180∘−−−−−(2)
From(1) and (2), we get∠4+∠6=∠2+∠4
∠6=∠2
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal.
Then, the lines are parallel.
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal.
Then, the lines are parallel.
Hence, the statement “If two lines intersected by a transversal form supplementary same-side interior angles, then the lines are parallel.” is proved.
Carnegie Learning Geometry Exercise 2.4 Student Solutions Page 103 Problem 20 Answer
Given that If two lines intersected by a transversal form supplementary same-side exterior angles, then the lines are parallel.
Also∠1+∠7=180∘ and ∠2+∠8=180∘
We have to prove statements for the Same-Side Exterior Angle Converse Conjecture.
We have∠1+∠7=180∘−−−−−(1) and ∠2+∠8=180∘−−−−−(2)
The angles∠6, ∠7 and ∠2,∠3 are vertically opposite angles. So they, are equal
∠6=∠7 and ∠2=∠3
We have∠4+∠6=180∘ −−−−−−−−(3) and ∠3+∠5=180∘
If a ray stands on a line, then the angles so formed is linear pair ∠2+∠4=180∘−−−−−(3)
From (3) and (4), we get
∠4+∠6=∠2+∠4
∠6=∠2
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal.
Then, the lines are parallel.
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal.
Then, the lines are parallel.
Hence, the statement “If two lines intersected by a transversal form supplementary same-side exterior angles, then the lines are parallel.” is proved.
Parallel and Perpendicular Lines Exercise 2.4 Carnegie Learning 2nd Edition Answers Page 103 Problem 21 Answer
We are given If two lines intersected by a transversal form supplementary same-side exterior angles, then the lines are parallel.
Given that∠1+∠7=180∘ and ∠2+∠8=180∘
We have to prove the Same-Side Exterior Angle Converse Conjecture.
We have∠4+∠6=180∘ −−−−−−−−(1) and ∠3+∠5=180∘
If a ray stands on a line, then the angles so formed is linear pair ∠2+∠4=180∘−−−−−(2)
From (1) and (2), we get
∠4+∠6=∠2+∠4
∠6=∠2
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal.
Then, the lines are parallel.
Thus, the lines are intersected by a transversal such that a pair of corresponding angles are equal.
Then, the lines are parallel.
Hence, the statement “If two lines intersected by a transversal form supplementary same-side exterior angles, then the lines are parallel.” is proved.