In the figure given below, p, q, r are parallel lines; l and m are two transversals. Consider the following: 1. AB : AC = DE : DF 2. AB × EF = BC × DE Which of the above is/are correct?
Both 1 and 2
Construct a perpendicular from A to the line CF, Let it meet BE at O and CF at P
Similarly a perpendicular from D is drawn to the line CF, let it meet BE and CF at R and S
In triangle ABO and ACP
∠A is common and ∠O = ∠P = 90°
ABO ∼ ACP
∴ AB/AC = AO/AP
In triangle DRE and DSF
∠D is common and ∠R = ∠S = 90°
DRE ∼ DSF
∴ DE/DF = DR/DS
Parallel lines are equally spaced
∴ AO = DR and AP = DS
∴ AB/AC = DE/DF
So statement 1 is correct
AB/ (AC - AB) = DE/ (DF - DE)
AB/BC = DE/EF
∴ AB × EF = BC × DE
So statement 2 is also correct
Similar Questions
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Important Questions from Lines and Angles
- Two parallel lines L1 and L2 are cut by a transversal. If one of the alternate interior angles is $65^\circ$, what is the measure of the corresponding alternate interior angle?
- Two lines AB and CD are parallel. Let PQ be the transversal intersecting AB at M and CD at N. If $\angle MND = 47^\circ$, find $\angle AMN$.
- The measure of an angle is one half of its supplementary angle. Find its measure.
- The sum of two angles is $155^\circ$ and their difference is $\frac{\pi}{2}$. The value of the greater angle (in radians) is:
- The measure of an angle is three times the measure of its complement. The angles are respectively:
