In trapezoid ABCD, the angle ABC is 30 degrees, the angle BCD is 135 degrees, and the side of CD is 29 cm.

| June 17, 2021 | education

In trapezoid ABCD, the angle ABC is 30 degrees, the angle BCD is 135 degrees, and the side of CD is 29 cm. Find the side of AB.

Given: a trapezoid, where ∠ABC = 30 °, ∠BCD = 135 °, CD = 29 cm.
It is required to determine: the length of the side AB.
Since the figure ABCD is a trapezoid, then AD || BC. Therefore, if we draw the height of the trapezoid EC ┴ BC, then ∠BCE = ∠CED = 90 °. Hence, ΔCED is a right-angled triangle.
Also, ∠DCE = ∠BCD – ∠BCE = 135 ° – 90 ° = 45 °, therefore ∠CDE = 90 ° – 45 ° = 45 °. Thus, ΔCED is an isosceles right triangle.
By the Pythagorean theorem, EC ^ 2 + ED ^^ 2 = CD2 or 2 * EC ^ 2 = CD ^ 2 = (29 cm) ^ 2, whence EC ^ 2 = (29 cm) ^ 2 / 2. Hence, EC = (29 cm) / (√2).
Let’s draw one more trapezoid height AF ┴ BC. Then, ∠AFB = 90 ° and ΔAFB is also a right triangle.
It is clear that AF = EC = (29 cm) / (√2).
Since ∠ABC = 30 °, then the following property of a right-angled triangle can be applied to ΔAFB: “The leg, which lies opposite an angle of 30 degrees, is equal to half of the hypotenuse.” We have AF = AB / 2.
Thus, AB / 2 = (29 cm) / (√2), whence AB = 2 * (29 cm) / (√2) = (√2) * 29 cm.
Answer: (√2) * 29 cm.



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