Which of the following statements is not true when two parallel lines are cut by a transversal?

Option 4 : The Sum of the interior angles formed on one side of the transversal line is a right angle

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Concept:

Corresponding Angles are angles that are in the same relative position at an intersection of a transversal and at least two lines. 

Alternate Angles: when a transverse line cuts two parallel lines, two angles lie on opposite sides of the transversal line and opposite relative sides of the other lines are called Alternate angles.

All angles that are either exterior angles, interior angles, alternate angles, or corresponding angles are all congruent.

Opposite Angles lie opposite to each other when two lines cross.

• Corresponding angles are always equal to each other.
• Alternate angles are always equal to each other.
• The sum of the interior angles formed on one side of the transversal line is two right angles.
• Opposite angles are also equal to each other.

Calculation:

In the above diagram, ‘l’ and ‘m’ are two parallel lines and ‘n’ is a transversal line that cuts both parallel lines.

Corresponding angles are ∠3 and ∠7,

∠4 and ∠8,

∠2 and ∠6,

∠1 and ∠5.

Alternate angles are ∠3 and ∠5,

∠4 and ∠6,

∠1 and ∠7,

∠2 and ∠8

Opposite angles are ∠2 and ∠4,

∠1 and ∠3

Hence, the sum of the interior angles formed on one side of the transversal line is 180° not a right angle.

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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 °

Co-interior angles are supplementary

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Alternate interior angles are equal.

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Corresponding angles are not equal

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There are four pair of Vertically opposite angles

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