What is the value of x? enter your answer in the box. x = cm a bow tie shape polygon made of two bow ties that share a vertex. the shape is created by two segments intersecting forming two triangles with vertical angles. an alternate interior angle of the triangles are marked congruent to each other. the larger triangle is on the right and the smaller triangle is on the left. on the smaller triangle, the north side is labeled 6 centimeters. the bottom side is labeled x centimeters. on the larger triangle, the north side is labeled 42 centimeters. the bottom side is labeled 36 centimeters.

 7 centimeters 

Explanation:

Let triangle ABC and triangle CDE form a bow tie such that 
     - triangle ABC is smaller than triangle CDE
     - Segment AC and segment CD are the north sides
     - Segment BC and segment CE are the bottom sides
     - AC = 6, CD = 42, CE = 36, BC = x

Since angle ACB and angle ECD are formed by intersecting segment AE and segment BD at point C, they are vertical angles. (See the attached picture) So, angle ACB and angle ECD are congruent. 

Moreover when we form our bowtie, point A is in the north side and point E is in the bottom sides and this implies that angle BAC and angle CED are alternate interior angles as shown in the attached figure. 

Since it is given in the problem that alternating interior angle are congruent, angle BAC and angle CED are congruent. 

Now, we have the following pairs of angles in triangle ABC and triangle CDE that are congruent:

- angle BAC and angle CED
- angle ACB and angle ECD

By AA Similarity theorem, when we have two pairs of congruent angles for two triangles, then the two triangles are similar. So, triangle ABC and triangle CDE are similar.

In similar triangles, the sides that are opposite to the congruent angles are proportional to each other. So, 

    (1)

Since , we can substitute these values to equation (1). So, equation (1) becomes


  (2)

SInce we are solving for x, we can multiply both sides of equation (2) by the denominator at the left side which is 42 so that

x = 42(1/6)
x = 7 centimeters

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