Questions: Find the range of values for (x). [2<x<[?]]

Find the range of values for (x). [2<x<[?]]
Transcript text: Find the range of values for $x$. \[ 2
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Solution

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Solution Steps

Step 1: Identify the given information

We are given an isosceles trapezoid with one pair of congruent sides marked. The lengths of the two parallel sides are 10 and 2x-4. Two angles of the trapezoid are given as 45° and 60°. A diagonal divides the trapezoid into two triangles.

Step 2: Analyze the triangles

The two congruent sides of the trapezoid have length 10. Since the sum of angles in a triangle is 180°, the third angle in the lower triangle is 180° - (60° + 45°) = 75°. The third angle in the upper triangle is 180° - (45° + (180° - 75°)) = 180° - 45° - 105° = 30°. We have two triangles. The lower triangle has angles 45°, 60°, and 75° and sides 10, let's say 'a', and 2x-4. The upper triangle has angles 30°, 75°, and 75°, with two equal sides of length 'a' and another side with length 10.

Step 3: Determine the range of x

In any triangle, the larger side is opposite the larger angle. In the lower triangle, since 75° > 60° > 45°, the sides opposite these angles follow the same inequality. So, 10 (opposite 75°) > a (opposite 60°) > 2x - 4 (opposite 45°). This means 2x - 4 < 10, which simplifies to 2x < 14, or x < 7. Also, in the upper triangle, since 75° > 75° > 30°, the sides opposite these angles follow the same inequality. Thus, a (opposite 75°) = a (opposite 75°) > 10 (opposite 30°). Since a > 10 and in the lower triangle we found a > 2x - 4, and the side with length 10 is the largest side, 2x-4 must be the smallest side. A side length must be greater than 0, so 2x - 4 > 0, or 2x > 4, which implies x > 2.

Final Answer

2 < x < 7

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