Theorem 6.2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
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To prove Theorem 6.2: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal, we will follow these steps:
### Step-by-Step Solution:
1. **Identify the Lines and Angles:**
- Let line AB and line CD be two parallel lines.
- Let line PQ be a transversal that intersects lines AB and CD at points X and Y, respectively.
- We need to prove that:
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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.
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.
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.
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.
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
A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.
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.
Alternate interior angles between parallel lines are equal
If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.
Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
