Question 23

Statements (I) and (II) are as given below:
I: If two lines intersect, then the vertically opposite angles are equal.
II: If a transversal intersects two other lines, then the sum of two interior angles on the same side of the transversal is $180^{\circ}$ .

Then,
(a) both (I) and (II) are true
(b) (I) is true and (II) is false
(c) (I) is false and (II) is true
(d) both (I) and (II) are false

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Solution

Statement (I)

If lines l and m intersect each other, then x and y are known as the vertically opposite angle. The vertically opposite angles so formed are equal.
$∴ x = y$
Statement (II)

If two lines l and m are intersected by a transversal P, then the sum of two interior angles will be $180^{\circ}$ , only if l and m are parallel.

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Similar questions

Which of the following statements are true (I) and which are false (F)? Give reasons.
(i) If two lines are intersected by a transversal, then corresponding angles are equal.
(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
(iii) Two lines perpendicular to the same line are perpendicular to each other.
(iv) Two lines parallel to the same line are parallel to each other.
(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.

Q. Write 'T' for true and 'F' for false

(i) If two parallel lines are cut by a transversal, then the alternate interior angles are equal.
(ii) If two lines intersect each other, then the vertically opposite angles are equal.
(iii) Each acute angle of an isosceles right triangle measures 60°.
(iv) A right triangle cannot have an obtuse angle.

Q. Read the following statements which are taken as axioms :
(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
(ii) If a transversal intersect two parallel lines, then alternate interior angles are equal.
Is this system of axioms consistent?

Fill in the blanks in each of the following to make the statement true:
(i) If two parallel lines are intersected by a transversal then each pair of corresponding angles are ___
(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are ___
(iii) Two lines perpendicular to the same line are ___ to each other.
(iv) Two lines parallel to the same line are to ___ each other.

(v) If a transversal intersects a pair of lines in such a way that a pair of alternate angles are equal, then the lines are ___
(vi) If a transversal intersects a pair of lines in such a way that the sum of interior angles on the same side of transversal is $180^{0}$ , then the lines are ___

Observe the following statements

$(i) I n Δ A B C, b c o s^{2} \frac{C}{2} + c c o s^{2} \frac{B}{2} = s$

$(i i) I n Δ A B C, c o t \frac{A}{2} = \frac{b + c}{a} \Rightarrow B = 90^{0}$

Which of the following is correct?

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