Alternate Interior Angles – Definitions, Properties, Theorem, Questions and Solutions

About Alternate Interior Angles

Alternate interior angles are angles that are created when two lines intersect inside a shape. The two lines create four angles, and the two angles that are opposite each other are the alternate interior angles.

What are Alternate Interior Angles?

Alternate interior angles are two angles that are adjacent to each other and have the same measure.

Alternate Interior Angles: Theorem and Proof Theorem

If two lines intersect, then the angle between them is the sum of the angles on the opposite side of the intersection.

Proof

Given:

AB and CD intersect at point P

Angles A and D are opposite each other

Angles B and C are opposite each other

Angles A and B are supplementary

Angles C and D are supplementary

Then:

Angle APD = Angle BPQ + Angle CQD

Antithesis of The Theorem

The antithesis of a theorem is a statement that is the exact opposite of the theorem. For example, the theorem that states that two points determine a line can be stated as the antithesis that states that two points do not determine a line

What is a Straight Angle?

A straight angle is an angle that is 180 degrees.

What are Parallel Lines?

Parallel lines are two lines that are always the same distance apart from each other.

Properties

• These angles are congruent.
• The sum of the angles formed on the same side of the transversal which are inside the two parallel lines is always equal to 180°.
• In the case of non – parallel lines, alternate interior angles don’t have any specific properties.

• Theorem and Proof

  Statement: The theorem states that “ if a transversal crosses the set of parallel lines, the alternate interior angles are congruent”.

  Given: a//d

  To prove: ∠4 = ∠5 and ∠3 = ∠6

  Proof: Suppose a and d are two parallel lines and l is the transversal that intersects a and d at points P and Q. See the figure given below.

   

  From the properties of the parallel line, we know if a transversal cuts any two parallel lines, the corresponding angles and vertically opposite angles are equal to each other. Therefore,

  ∠2 = ∠5 ………..(i) [Corresponding angles]

  ∠2 = ∠4 ………..(ii) [Vertically opposite angles]

  From eq.(i) and (ii), we get;

  ∠4 = ∠5 [Alternate interior angles]

  Similarly,

  ∠3 = ∠6

  Hence, it is proved.
  

  

  Antithesis of Theorem

  If the alternate interior angles produced by the transversal line on two coplanar are congruent, then the two lines are parallel to each other.

  Given: ∠4 = ∠5 and ∠3 = ∠6

  To prove: a//b

  Proof: Since ∠2 = ∠4 [Vertically opposite angles]

  So, we can write,

  ∠2 = ∠5, which are corresponding angles.

  Therefore, a is parallel to b.
  

  Co-interior Angles

  Co-interior angles or Consecutive interior angles are the two angles that are on the same side of the transversal. Co-interior angles are the interior angles and it sums up to 180 degrees. It means that the sum of two interior angles, which are on the same side of transversal is supplementary. Co-interior angles resemble like in “C” shape and both the angles are not equal to each other. The co-interior angle is also known as the consecutive interior angles or the same side interior angles.
  

  Co-interior Angle Theorem and Proof

  Statement:
  If the transversal intersects the two parallel lines, each pair of co-interior angles sums up to 180 degrees (supplementary angles).

  Proof:

   

  Let us consider the image given above:

  In the figure, angles 3 and 5 are the co interior angles and angles 4 and 6 are the co-interior angles.

  To prove: ∠3 and ∠5 are supplementary and ∠4 and ∠6 are supplementary.

  Given that, a and b are parallel to each other and t is the transversal.

  By the definition of linear pair,

  ∠1 and ∠3 forms the linear pair.

  Similarly, ∠2 and ∠4 form the linear pair.

  By using the supplement postulate,

  ∠1 and ∠3 are supplementary

  (i.e.) ∠1 + ∠3 = 180

  Similarly,

  ∠2 and ∠4 are supplementary

  (i.e.) ∠2 + ∠4 = 180

  By using the corresponding angles theorem, we can write

  ∠1 ≅∠5 and ∠2 ≅ ∠6

  Thus, by using the substitution property, we can say,

  ∠3 and ∠5 are supplementary and ∠4 and ∠6 are supplementary.

  Hence, the co-interior angle theorem (consecutive interior angle) is proved.

  The converse of this theorem is “if a transversal intersects two lines, such that the pair of co-interior angles are supplementary, then the two lines are parallel”.

  Facts about consecutive interior or co-interior angles:

  • Consecutive interior or co-interior angles contains different vertices but lie on the same side of the transversal
  • These angles lie between two lines
  • Consecutive interior angles are non-adjacent angles
  • If a transversal is drawn on two parallel lines, then the sum of co-interior angles formed are always added up to 180 degrees
  • The sum of consecutive interior angles of a parallelogram is always supplementary
  

  Examples

  Question 1:

  Find the value of B and D in the given figure.

   

  Solution:

  Since 45° and D are alternate interior angles, they are congruent.

  So, D = 45°

  Since 135° and B are alternate interior angles, they are congruent.

  So, B = 135°

  Question 2:

  Find the missing angles A, C and D in the following figure.

   

  Solution:

  As angles ∠A, 110°, ∠C and ∠D are all alternate interior angles, therefore;

  ∠C = 110°

  By supplementary angles theorem, we know;

  ∠C+∠D = 180°

  ∠D = 180° – ∠C = 180° – 110° = 70°