Same side interior angles are two angles that are on the interior of (between) the two lines and specifically on the same side of the transversal. The same-side interior angles sum up to 180 degrees. When two parallel lines are intersected by a transversal line they formed 4 interior angles. The 2 non-adjacent interior angles that are on the same side of the transversal are supplementary. Show
What are Same Side Interior Angles?When two parallel lines are intersected by a transversal, 8 angles are formed. The same side interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal. So the same side interior angles:
The "same side interior angles" are also known as "co-interior angles." The 8 angles thus formed are classified into different types of angles listed below: In the given figure, line AB || CD and line l is the transversal. From the "Same Side Interior Angles - Definition," the pairs of same side interior angles in the above figure are: Same Side Interior Angles TheoremLet us consider the above figure. In the above figure, lines AB and CD are parallel and L is the transversal. We just read that the pairs of the same side interior angles in the above figure are: The relation between the same side interior angles is determined by the same side interior angle theorem. The theorem for the "same side interior angle theorem" states: If a transversal intersects two parallel lines, each pair of same-side interior angles are supplementary (their sum is 180°). Same Side Interior Angles Theorem ProofReferring the above figure once again: ∠4 = ∠8, and ∠3 = ∠7 [corresponding angles are equal]. From the above two equations, ∠4 + ∠5 = 180° Similarly, ∠3 + ∠6 = 180° Hence proved, that each pair of same-side interior angles are supplementary. Converse of Same Side Interior Angles TheoremThe converse of the same-side interior angle theorem states that if a transversal intersects two lines such that a pair of same-side interior angles are supplementary, then the two lines are parallel. Converse of Same Side Interior Angles Theorem ProofConsidering same above figure, ∠4 + ∠5 = 180° ⇒ (1) Since ∠5 and ∠8 forms linear pair, ∠5 + ∠8 = 180° ⇒ (2) From (1) and (2), ∠4 = ∠8 Thus, a pair of corresponding angles are equal, which can only happen if the two lines are parallel. Hence, the converse of the same side interior angle theorem is proved. Important Notes The following are the important points related to the same side interior angles.
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FAQs on Same Side Interior AnglesThe same side interior angles are NOT congruent. They are supplementary. The same side interior angles formed when two parallel lines intersected by a transversal. The same side interior angles can be congruent only when each angle is equal to a 90 degree because then the sum of the same side interior angles is equal to 180 degrees. Are Same Side Interior Angles Adjacent?The same side interior angles are always non-adjacent because the angles are formed on the two different lines that are parallel to each other. What is the Sum of the Two Same Side Interior Angles on the Transversal?When two parallel lines crossed by a transversal they formed same-side interior angles and their sum is equal to 180 degrees. As the sum of the same side interior angles is 180 degrees therefore the angles are supplementary. What is the Converse of Same Side Interior Angles?The converse of the same-side interior angle states that when two lines intersected by a transversal and the angles inside on the same side are supplementary or we can say the sum of inside angles on the same side is 180 degrees then the lines are said to be parallel. What is Another Name of the Same Side Interior Angles?The same side interior angles are also known as consecutive interior angles as the angles are on one side of the transversal but inside the two parallel lines. What is the Difference Between Same Side Interior Angles and Same Side Exterior Angles?When two parallel lines are intersected by a transversal line 8 angles were formed. The same side interior angles are the angles inside the parallel lines on the same side of the transversal and the same side exterior angles are the angles outside the parallel lines on the same side of the transversal. What is the Difference Between the Same Side Interior Angles and Corresponding Angles?The difference between the same side interior angles and corresponding angles is corresponding angles are congruent whereas, in the case of the same side of interior angles, the sum of the same side interior angles is equal to 180 degrees only if the transversal line intersects two parallel lines.
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 ° |