If two lines are parallel, then the corresponding angles formed by a transversal are equal.
Step-by-step solution
Idea: The proposition is the corresponding angles property of parallel lines. For the converse, suppose the lines were not parallel: they would meet and form a triangle, and the exterior angle property would make the two corresponding angles unequal.
- Converse: If the corresponding angles formed by a transversal (cutting two lines) are equal, then the lines are parallel.½ mark
- Proposition — true. This is the corresponding angles axiom for parallel lines: when a transversal cuts two parallel lines, each pair of corresponding angles is equal.½ mark
- Converse — true. Let a transversal t cut lines l and m at P and Q, with corresponding angles ∠1 = ∠2 (see the diagram). Suppose l and m are not parallel. Then they meet at some point R, forming △PQR.½ mark
- ∠1 is an exterior angle of △PQR at P, and ∠2 (= ∠PQR) is one of its interior opposite angles. By the exterior angle property, ∠1 = ∠2 + ∠R.½ mark
- ∠R is more than 0°, so ∠1 > ∠2. This contradicts ∠1 = ∠2. So l and m cannot meet: they are parallel. (If they met on the other side, the same argument works with the angles on that side.)½ mark
- So the proposition and its converse are both true.½ mark
Answer to write in the exam
Converse: If the corresponding angles formed by a transversal are equal, then the lines are parallel.
Proposition: True (corresponding angles axiom).
Converse: Let ∠1 = ∠2 (corresponding angles at P and Q). Suppose l and m meet at R.
In △PQR, ∠1 is an exterior angle and ∠2 an interior opposite angle.
∠1 = ∠2 + ∠R > ∠2 (exterior angle property) — contradiction.
∴ l ∥ m; the converse is true.
∴ Both the proposition and its converse are true.
Common mistakes that cost marks
- Mixing up corresponding angles with alternate or co-interior angles.
- Saying the converse is true ‘because the proposition is true’, with no reason.
- Using the parallel lines property inside the proof of the converse — we do not yet know the lines are parallel.
How this can come in the exam
A transversal cuts lines l and m, and one pair of corresponding angles are both 72°. Which is correct?
- l and m meet on the right
- l ∥ m
- l ⊥ m
- Nothing can be said
Show answer
(B) l ∥ m
Equal corresponding angles mean the lines are parallel (the converse proved above).
Assertion (A): If a transversal makes corresponding angles of 65° and 70° with two lines, the lines are not parallel.
Reason (R): If two lines are parallel, then the corresponding angles formed by a transversal are equal.
- Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true but R is not the correct explanation of A.
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
Show answer
(A) Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
If the lines were parallel, R would force the corresponding angles to be equal; 65° ≠ 70°, so they are not parallel. R explains A.
Try one yourself
Two lines are cut by a transversal so that a pair of corresponding angles are (3x + 10)° and (5x − 30)°. For what value of x are the lines parallel?
Show answer
Equal corresponding angles: 3x + 10 = 5x − 30, so 2x = 40 and x = 20 (both angles are 70°).
More questions like this
- If a quadrilateral is a square, then all its angles are equal.
- Given any ∆ABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle.
Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown.
Proposition: If AB = AC, then IE = IF. - If x = y, then a + x = a + y, where x, y and a are any three numbers.
This proposition and its converse are routinely used while solving equations. - If a and b are perfect squares, then ab is a perfect square.
- If x = y, then x2 = y2.