Consecutive interior angles are a pair of angles that lie between two lines and on the same side of a transversal. When the two lines are parallel, their sum is 180°, making them supplementary angles.
Example 1: If a transversal cuts two parallel lines and a pair of consecutive interior angles measure (4x + 8)° and (16x + 12)°, calculate the value of x and the value of both consecutive interior angles.
Solution:
Because the supplied lines are parallel, the inner angles (4x + 8)° and (16x + 12)° are consecutive. These angles are additional according to the consecutive interior angle theorem.
As a result, (4x + 8)° + (16x + 12)° = 180°
⇒ 20x + 20 = 180°
⇒ 20x = 180° - 20°
⇒ 20x = 160°
⇒ x = 8°
Let us now substitute x for the values of the subsequent interior angles.
Thus, 4x + 8 = 4(8) + 8 = 40° and
16x + 12 = 16(8) + 12 = 140°
Thus, value of both consecutive interior angles 40° and 140°.
Example 2: The value of ∠3 is 85°, and ∠6 is 110°. Now, check that the 'n' and 'm' lines are parallel.
Solution:
If the angles 110° and 85° in the above figure are supplementary, then the lines 'n' and 'm' are parallel.
However, 110° + 85° = 195°, indicating that 110° and 85° are NOT supplementary.
As a result, the given lines are NOT parallel, according to the Consecutive Interior Angles Theorem.
Example 3: Find the missing angles ∠3, ∠5, and ∠6. In the diagram, ∠4 = 65°.
Solution:
Given: ∠4 = 65°, ∠4 and ∠6 are corresponding angles, therefore;
∠6 = 65°
By supplementary angles theorem, we know;
∠5 + ∠6 = 180°
∠5 = 180° – ∠6 = 180° – 65° = 115°
Since,
∠3 = ∠6
Therefore, ∠3 = 115°.
Practice Problems
Problem 1: In a pair of parallel lines cut by a transversal, if one co-interior angle measures (2x - 7)° and other is (x + 1)°, then what is the measure of both co-interior angles?
Problem 2: If angle P is a co-interior angle with angle Q on a pair of parallel lines, and angle Q measures 60°, what is the measure of angle P?
Problem 3: In a pair of parallel lines intersected by a transversal, if sum of both cosecutive interior angles is (3z-8)° and one of the co-interior angle is z. Then find the value of both cosecutive interior angles.
