two parellel lines p and m are intersected by a transversal t. if the interior angles on the same side of transveral are (2 x- 8) and (3 x -7) find the measure of these angles.
Answers
Step-by-step explanation:
Given: Two parallel lines AB and CD and a transversal EF intersect them at G and H respectively. GM, HM, GL and HL are the bisectors of the two pairs of interior angles.
To Prove: GMHL is a rectangle.
Proof:
∵AB∥CD
∴∠AGH=∠DHG (Alternate interior angles)
⇒
2
1
∠AGH=
2
1
∠DHG
⇒∠1=∠2
(GM & HL are bisectors of ∠AGH and ∠DHG respectively)
⇒GM∥HL
(∠1 and ∠2 from a pair of alternate interior angles and are equal)
Similarly, GL∥MH
So, GMHL is a parallelogram.
∵AB∥CD
∴∠BGH+∠DHG=180
o
(Sum of interior angles on the same side of the transversal =180
o
)
⇒
2
1
∠BGH+
2
1
∠DHG=90
o
⇒∠3+∠2=90
o
.....(3)
(GL & HL are bisectors of ∠BGH and ∠DHG respectively).
In ΔGLH,∠2+∠3+∠L=180
o
⇒90
o
+∠L=180
o
Using (3)
⇒∠L=180
o
−90
o
⇒∠L=90
o
Thus, in parallelogram GMHL, ∠L=90
o
Hence, GMHL is a rectangle.