In the diagram, is an altitude drawn in trapezoid DEFG, DE = 10, EF = 9, FG = 8, and GD = 15. What is mÐD to the nearest degree?

37
53
60
80

To find the measure of angle D, we can use trigonometry and the properties of right triangles.

First, we need to draw a diagram to visualize the trapezoid DEFG:

```
F ________ G
| /
| /
h | /
| /
| /
| /
| /
|/
D________E

```

In the diagram, h represents the length of the altitude DE.

To find the measure of angle D, we can use the tangent function. The tangent of angle D is defined as the length of the side opposite angle D (h) divided by the length of the side adjacent to angle D (GD).

Therefore, tan(D) = h / GD

Plugging in the known values, tan(D) = h / 15

Now, solving for D:

D = arctan(h / 15)

To find h, we can use the Pythagorean theorem. In triangle GDE, GD is the hypotenuse and DE and h are the legs.

By Pythagoras theorem, we have:

DE^2 + h^2 = GD^2

Substituting the given values:

10^2 + h^2 = 15^2

Simplifying the equation:

100 + h^2 = 225

h^2 = 225 - 100

h^2 = 125

Taking the square root of both sides:

h = √125

h = 11.18 (rounded to two decimal places)

Now we can substitute the value of h into our equation for D:

D = arctan(11.18 / 15)

Using a calculator, we can find:

D ≈ 37 degrees

Therefore, the measure of angle D to the nearest degree is 37.

To find the measure of angle D in trapezoid DEFG, we need to use the properties of a trapezoid.

In a trapezoid, the opposite angles are supplementary, which means they add up to 180 degrees. Since angle F and angle G are opposite, we have:
m∠F + m∠G = 180°

Next, we need to find the measure of angle F. In triangle DEF, angle F is a right angle because DE is an altitude. Therefore, m∠F = 90°.

Using the equation above, we can solve for m∠G:
90° + m∠G = 180°
m∠G = 180° - 90°
m∠G = 90°

Now, to find the measure of angle D, we use the fact that the sum of the angles in a triangle is 180 degrees. Since triangle DGF is a right triangle, the sum of angles D, G, and F must be 180 degrees.

m∠D + m∠G + m∠F = 180°
m∠D + 90° + 90° = 180°
m∠D + 180° = 180°
m∠D = 0°

Therefore, the measure of angle D is 0 degrees to the nearest degree.

60