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Two angles, formed when a line crosses two other lines, that lie on opposite sides of the transversal line and on opposite relative sides of the other lines. If the two lines crossed are parallel, the alternate angles are equal. Alternate interior angles: Alternate interior angles are the pair of angles lying in the region between the two lines (intersected by a transversal) and on the opposite sides of the transversal but one below the transversal and the other above the transversal. In the adjoining figure and are pair of alternate interior angles.Theorem: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. Given: m and n are parallel lines and the transversal l cuts m and n To Prove : Proof: Corresponding angles are equal as m and n are parallel lines and l is the transversal
Vertically opposite angles are equal when m and l intersect.
From Eq. (1) and (2)
Hence Proved that alternate interior angles are equal. Theorem: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the lines are parallel. Given : m and n are lines and the transversal l cuts m and n and To Prove : m and n are parallel lines Proof: Vertically opposite angles are equal when m and l intersect.
and From Eq. (1) and (2)
But they are Corresponding angles as m and n are lines and l is the transversal and as they are equal the lines are parallel. Hence Proved that m and n are parallel lines. Same is the case with exterior alternate angles Alternate exterior angles: Alternate exterior angles are the pair of angles lying outside the region between the two lines (intersected by a transversal) and on the opposite sides of the transversal but one below the transversal and the other above the transversal. In the given figure and are pair of alternate exterior angles. |
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Illustration: Find the angle which is alternate to :
Solution:
Illustration: In the given figure m || n and = 5:3. Find the value of . Solution: Let the common factor be x |
Illustration: In the given figure, PQ and RS are two mirrors placed parallel to each other . An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.
Solution:
Construction: Draw BB' PQ and CC' RS.
Since BB' and CC' are normals, therefore and [Angle of incidence = angle of reflection]
Since PQ || RS, therefore BB' || CC'
Now, BB' || CC' and BC is a transversal
Multiply by 2 on both sides, we have
But, are alternate angles formed by transversal BC with AB and CD.
Now, using the theorem "If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the lines are parallel".
Hence, AB || CD.
Hence the proof.