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.
Text Solution
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Since ray `ND` stands on line `EF`,
`/_3 + /_5=180 ` (linear pair of angles)
But, `/_ 1 = /_5` (Corresponding angles as `AB∣∣CD`)
From above equations, we get
`/_ 1+ /_ 3=180`
Again ray `CN` stands on `EF`,
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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.
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 6.2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is 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.
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.
If the bisectors of a pair of alternate angles formed by a transversal with two given lines are parallel, prove that the given lines are parallel.
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.
Two straight lines are cut by a transversal. If the bisectors of a pair of co-interior angles are perpendicular to each other , prove the two straight lines are parallel to each other.
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 one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
