the longer base of an isosceles trapezoid is 14 feet. the nonparallel sides are each 10 feet long and the length of diagonal is 16 feet. find the measure of a base angle

So we have a trapezoid ABCD, in which

AB=unknown
BC=10,
CD=14
DA=10
and
AC=16.

Sketch the triangle ACD for which the lengths of all three sides are known.
Using the cosine rule, calculate the cosine of A using
cosA=(AC²+AD²-CD²)/(2*AC*AD)
and hence calculate ∠A.

To find the measure of a base angle in an isosceles trapezoid, we can use the properties of isosceles trapezoids and the given information.

An isosceles trapezoid is a quadrilateral with one pair of opposite sides that are parallel. It also has two nonparallel sides of equal length. Let's denote the longer base as "a" with a length of 14 feet and the nonparallel sides as "b" with a length of 10 feet each. The length of the diagonal is denoted as "d" and is given as 16 feet.

First, we can draw the isosceles trapezoid and label the given measurements:
```
b b
________a_______
\ /
\ /
\ /
\ /
d \ / d
\ /
\ /
\/
```
Given:
a (longer base) = 14 feet
b (nonparallel sides) = 10 feet
d (diagonal) = 16 feet

Now, let's analyze the properties of isosceles trapezoids. Since it is an isosceles trapezoid, the two base angles are congruent (equal). Let's denote one of the base angles as "x."

To find the measure of an angle in a triangle, we can use the law of cosines. In this case, we can form a triangle using one of the nonparallel sides, the diagonal, and one of the base angles.

In the triangle, the two known sides are:
Side a (longer base) = 14 feet
Side b (nonparallel side) = 10 feet

The diagonal, d, acts as the side opposite to the angle we want to find, x.

Now, let's apply the law of cosines:

cos(x) = (a^2 + b^2 - d^2) / (2 * a * b)

Substituting the given values:
cos(x) = (14^2 + 10^2 - 16^2) / (2 * 14 * 10)

Calculating:
cos(x) = (196 + 100 - 256) / (280)
cos(x) = 40 / 280
cos(x) = 0.14285714285714285

Now, we need to find the value of x by taking the inverse cosine (cos^-1) of 0.14285714285714285.

Using a calculator, we can find:
x ≈ 82.15 degrees

Therefore, the measure of a base angle in the given isosceles trapezoid is approximately 82.15 degrees.