Parallel Lines and Angle Calculation
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22. $a \parallel b$, $c \parallel d$, $e, f$ düz xətləri kəsən, $\alpha = 2x+3y$, $\alpha > 90^\circ$ olarsa, $(x+y)$ cəminin ən kiçik qiymətini tapın $(x, y \in N)$.
A) $45^\circ$ B) $30^\circ$ C) $50^\circ$ D) $35^\circ$ E) $40^\circ$
This question includes visual content: The image shows a geometry problem involving two pairs of parallel lines: 'a' is parallel to 'b' (a || b) and 'c' is parallel to 'd' (c || d). Two transversals 'e' and 'f' intersect these lines. The diagram displays four labeled angles: an angle 'alpha' on transversal 'e', an angle marked (4x-y) degrees near line 'c', an angle marked (7x-3y) degrees near line 'd', and an angle marked (3x+2y) degrees near line 'b' intersected by transversal 'f'.
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Step by Step Written Solution
Hi Ayse, let's solve this geometry problem together. We are given parallel lines a and b, as well as c and d. We need to find the minimum integer sum of x and y, given the angle conditions provided.
Analyzing the Geometry
First, observe lines c and d are parallel. The angles labeled (4x-y) degrees and (7x-3y) degrees are interior angles on the same side of the transversal. Their sum must be 180 degrees.
Let's simplify this equation. Combining the x and y terms gives us 11x minus 4y equals 180.
We are also given alpha equals 2x plus 3y, and alpha must be greater than 90 degrees. Let's express y in terms of x using our main equation.
Since x and y are natural numbers, 11x minus 180 must be divisible by 4. This means 11x must be a multiple of 4, so x must be a multiple of 4. Let x equal 4k.
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