Theorem 6.4 : If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
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To prove Theorem 6.4: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary, we will follow these steps:
### Step-by-Step Solution:
1. **Identify the Parallel Lines and Transversal**:
- Let the two parallel lines be \( AB \) and \( CD \).
- Let the transversal be \( PQ \) that intersects \( AB \) at point \( M \) and \( CD \) at point \( N \).
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Theorem 6.5 : If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
Theorem 6.2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
Theorem 6.3 : If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
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...
Which of the following statements are true (T) 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 angle on the same side of the transversal are equal.
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 ? Justify your answer.
A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.
If two parallel lines are intersected by a transversal, prove that the bisectors of the interior angles on the same side of transversal intersect each other at right angles.
In the adjoining figure, identify(i) the pairs of corresponding angles.(ii) the pairs of alternate interior angles.(iii) the pairs of interior angles on the sameside of the transversal.(iv) the vertically opposite angles.
If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2:3, then the measure of the larger angle is (a) 54^0 (b) 120^0 (c) 108^0 (d) 136^0
