Which of the following statements is false a when a transversal cuts two lines such that pairs of corresponding angles are equal then the lines have to be parallel?

Maths-

Explanation :-

alternate interior angles are supplementary

False, alternate interior angles are equal but not supplementary.
Therefore, Option A is correct

Maths-General

Explanation :-

alternate interior angles are supplementary

False, alternate interior angles are equal but not supplementary.
Therefore, Option A is correct

Corresponding angles are equal
Alternate angles are equal
Sum of the interior angles formed on one side of the transversal line is two right angles
The Sum of the interior angles formed on one side of the transversal line is a right angle

In geometry, a transversal is a line that intersects two or more other (often parallel ) lines.

In the figure below, line n is a transversal cutting lines l and m .

When two or more lines are cut by a transversal, the angles which occupy the same relative position are called corresponding angles .

In the figure the pairs of corresponding angles are:

∠ 1 and ∠ 5 ∠ 2 and ∠ 6 ∠ 3 and ∠ 7 ∠ 4 and ∠ 8

When the lines are parallel, the corresponding angles are congruent .

When two lines are cut by a transversal, the pairs of angles on one side of the transversal and inside the two lines are called the consecutive interior angles .

In the above figure, the consecutive interior angles are:

∠ 3 and ∠ 6 ∠ 4 and ∠ 5

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary .

When two lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called the alternate interior angles .

In the above figure, the alternate interior angles are:

∠ 3 and ∠ 5 ∠ 4 and ∠ 6

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

When two lines are cut by a transversal, the pairs of angles on either side of the transversal and outside the two lines are called the alternate exterior angles .

In the above figure, the alternate exterior angles are:

∠ 2 and ∠ 8 ∠ 1 and ∠ 7

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

Example 1:

In the above diagram, the lines j and k are cut by the transversal l . The angles ∠ c and ∠ e are…

A. Corresponding Angles

B. Consecutive Interior Angles

C. Alternate Interior Angles

D. Alternate Exterior Angles

The angles ∠ c and ∠ e lie on either side of the transversal l and inside the two lines j and k .

Therefore, they are alternate interior angles.

The correct choice is C .

Example 2:

In the above figure if lines A B ↔ and C D ↔ are parallel and m ∠ A X F = 140 ° then what is the measure of ∠ C Y E ?

The angles ∠ A X F and ∠ C Y E lie on one side of the transversal E F ↔ and inside the two lines A B ↔ and C D ↔ . So, they are consecutive interior angles.

Since the lines A B ↔ and C D ↔ are parallel, by the consecutive interior angles theorem , ∠ A X F and ∠ C Y E are supplementary.

That is, m ∠ A X F + m ∠ C Y E = 180 ° .

But, m ∠ A X F = 140 ° .

Substitute and solve.

140 ° + m ∠ C Y E = 180 ° 140 ° + m ∠ C Y E − 140 ° = 180 ° − 140 ° m ∠ C Y E = 40 °

Answer

Verified

Hint: According to the question given in the question we have to determine that the statement If two parallel lines are cut by transversal, then a pair of alternate interior angles not equal is true or false. So, first of all we have to draw two parallel lines and a transversal line which intersects the two parallel lines which is as below:

Hence, from in the diagram above lines P and Q are parallel to each other and a transversal line R intersects the both of the parallel lines P and Q and all the interior angles are $\angle 1,\angle 2,\angle 3,\angle 4,\angle 5,\angle 6,\angle 7,$ and $\angle 8$.Now, we have to use the transversal theorem to determine when two parallel lines are cut by transversal, then pair of alternate interior angles not equal is true or not which is explained below:According to transversal theorem if two parallel lines are intersected by another transversal line then the alternate interior angles formed are congruent or we can say equal to each other.

Complete step-by-step solution:

Step 1: First of all we have to draw the diagram for If two parallel lines are cut by transversal, then a pair of alternate interior angles not equal is true or false. So, first of all we have to draw two parallel lines and a transversal line which intersects the two parallel lines as mentioned in the solution hint. Hence,

Step 2: Now, as we know that when two lines are intersected by a transversal line, then pairs of angles on one side of the transversal line which is R and inside the two lines P and Q are known as the consecutive interior angles. Hence, from the diagram as mentioned in the solution step 1,

$ \Rightarrow \angle 1 = \angle 5, \\ \Rightarrow \angle 3 = \angle 7, \\ \Rightarrow \angle 2 = \angle 6,and \\ \Rightarrow \angle 4 = \angle 8 $Hence, If two parallel lines are cut by transversal, then a pair of alternate interior angles not equal is false.

Therefore option (B) is correct.

Note: If two parallel lines are intersected by another transversal line then the alternate interior angles formed are congruent or we can say equal to each other.

When a transversal cuts two lines such that pairs of alternate interior angles are , then the lines have to be parallel.

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