Parallel lines and transversals
Find unknown angles in diagrams involving parallel lines and transversals using corresponding, alternate and co-interior angle properties.
Worked examples
Using co-interior angles
Straightforward
Problem
Two parallel lines are cut by a transversal. One co-interior angle measures . Find the other co-interior angle.
1
Identify the relationship. Co-interior angles lie between the parallel lines on the same side of the transversal.
The two angles are co-interior (also called same-side interior or allied angles).
2
State the co-interior angle property.
Co-interior angles are supplementary: they add to .
3
Find the unknown angle.
Answer
Two-step angle problem with parallel lines
Moderate
Problem
Two parallel lines are cut by a transversal. At the upper intersection, angle and lie on a straight line. Find the alternate interior angle at the lower intersection.
1
Find angle using angles on a straight line.
2
Apply the alternate interior angles property.
Alternate interior angles are equal when lines are parallel, so .
Answer
Solving an algebraic equation using co-interior angles
Challenging
Problem
Two parallel lines are cut by a transversal. The co-interior angles are and . Find and the size of each angle.
1
Write an equation using the co-interior angle property.
2
Collect like terms.
3
Solve for .
4
Find the size of each angle.
and
5
Check: co-interior angles must sum to .
✓
Answer
; the angles are and
Practise
Q1·Straightforward
Two parallel lines are cut by a transversal. One of the angles formed at the first intersection is . Find the size of the corresponding angle at the second intersection.
Explanation
Corresponding angles are equal when formed by a transversal crossing parallel lines. The corresponding angle is also .
Q2·Straightforward
Two parallel lines are cut by a transversal. An interior angle at one intersection measures . Find the size of the alternate interior angle at the other intersection.
Explanation
Alternate interior angles are equal when formed by a transversal crossing parallel lines. The alternate interior angle is also .
Q3·Straightforward
Two parallel lines are cut by a transversal. One co-interior angle measures . Find the other co-interior angle.
Explanation
Co-interior angles are supplementary: .
Q4·Straightforward
Two parallel lines are cut by a transversal. One co-interior angle measures . Find the other co-interior angle.
Explanation
Co-interior angles are supplementary: .
Q5·Moderate
Two parallel lines are cut by a transversal. At the first intersection, angles and lie on a straight line. Angle at the second intersection corresponds to angle . Find .
Explanation
Step 1: Angles on a straight line sum to , so . Step 2: Corresponding angles are equal, so .
Q6·Moderate
Two parallel lines are cut by a transversal. At the upper intersection, angle and lie on a straight line. Angle at the lower intersection is the alternate interior angle to . Find .
Explanation
Step 1: (angles on a straight line). Step 2: Alternate interior angles are equal, so .
Q7·Moderate
Two parallel lines are cut by a transversal. Angle is vertically opposite to a angle at the upper intersection. Angle at the lower intersection is co-interior to . Find .
Explanation
Step 1: Vertically opposite angles are equal, so . Step 2: Co-interior angles sum to , so .
Q8·Moderate
Two parallel lines are cut by a transversal. Angle at the lower intersection corresponds to a angle at the upper intersection. Angle is supplementary to . Find .
Explanation
Step 1: Corresponding angles are equal, so . Step 2: Supplementary angles sum to , so .
Q9·Challenging
Two parallel lines are cut by a transversal. The co-interior angles are and . Find the value of .
Explanation
Co-interior angles sum to : . Simplify: , so and . Check: and ; sum ✓
Q10·Challenging
Two parallel lines are cut by a transversal. The co-interior angles are and . Find the value of .
Explanation
Co-interior angles sum to : . Simplify: , so and . Check: and ; sum ✓
Q11·Challenging
Two parallel lines are cut by a transversal. The alternate interior angles are and . Find the value of .
Explanation
Alternate interior angles are equal: . Subtract : . Add : , so . Check: and ✓
Q12·Challenging
Two parallel lines are cut by a transversal. The corresponding angles are and . Find the value of .
Explanation
Corresponding angles are equal: . Subtract : . Add : , so . Check: and ✓