The isosceles trapezoid is part of an isosceles triangle with a 38° vertex angle. Also a acute base angle that measures 71. What is the measure of an obtuse base angle of the trapezoid? The diagram is not drawn to scale.
Let's label the trapezoid as ABCD, with AB as the longer base and DC as the shorter base. Let angle A be the obtuse base angle of the trapezoid.
We know that an isosceles trapezoid has one pair of parallel sides and two pairs of congruent angles.
Based on the given information, we can see that angle ACD is a right angle since it is part of a triangle with a vertex angle of 38°. Therefore, angle BCD is also a right angle since the opposite angles in a trapezoid are equal.
Let's find the third angle in triangle BCD. The sum of all angles in a triangle is 180°, so angle BCD + angle BDC + angle CBD = 180°. Since angle BCD and angle BDC are right angles, angle CBD must be 180° - 90° - 90° = 0°. This means that line segment BC is parallel to line segment AD.
Since line segment BC is parallel to line segment AD, the opposite angles of AD are also equal to the opposite angles of BC. Therefore, angle DAB is congruent to angle BCD, which is a right angle.
Let's find angle BAC, which is one of the acute base angles of the trapezoid. Since angle DAB is congruent to angle BCD, angle BAC is also congruent to angle BCD. Therefore, angle BAC is a right angle, meaning that it is 90°.
Now, let's find angle CAD, which is the difference between angle BAC (90°) and angle CAD (71°). Therefore, angle CAD = 90° - 71° = 19°.
Since angle CAD is the difference between angle BAC (90°) and angle CAD (71°), we can conclude that angle CAD is greater than 90°. Therefore, angle A is an obtuse angle.
Hence, the measure of angle A, the obtuse base angle of the trapezoid, is greater than 90°.