Find the smaller angle of the isosceles trapezoid ABCD if the diagonal AC makes 30 ° and 105

Find the smaller angle of the isosceles trapezoid ABCD if the diagonal AC makes 30 ° and 105 ° angles with the base BC and the lateral side CD, respectively.

Consider an isosceles trapezoid ABCD, where

AB = CD, ACB = 30 °, ACD = 105 °.

Note that the angles ACB and CAD are crosswise. Hence,

CAD = ACB = 30 °.

Now consider the triangle ACD.

In this triangle, we know 2 angles:

CAD = 30 °, ACD = 105 °. Then the third corner of this triangle is:

CDA = 180 ° – CAD – ACD = 180 ° – 30 ° – 105 ° = 45 °.

Since the trapezoid ABCD is isosceles, then

BAD = CDA = 45 °,

ABC = BCD = ACB + ACD = 30 ° + 105 ° = 135 °.

Answer: The smaller trapezoid angle ABCD is 45 °.