Angle D O E contains points D Left parenthesis 2 comma 3 right parenthesis, O Left parenthesis 0 comma 0 right parenthesis, and E Left parenthesis 5 comma 1 right parenthesis. Which of the following angles are both adjacent and supplementary to Angle D O E?
(1 point)
Responses
Angle D O F, with F Left parenthesis 10 comma 2 right parenthesis
Image with alt text: Angle D O F , with F Image with alt text: Left parenthesis 10 comma 2 right parenthesis
Angle D O F, with FLeft parenthesis negative 2 comma negative 3 right parenthesis
Image with alt text: Angle D O F , with F Image with alt text: Left parenthesis negative 2 comma negative 3 right parenthesis
Angle D O F, with FLeft parenthesis negative 5 comma negative 1 right parenthesis
Image with alt text: Angle D O F , with F Image with alt text: Left parenthesis negative 5 comma negative 1 right parenthesis
Angle E O F, with FLeft parenthesis negative 5 comma negative 1 right parenthesis
Image with alt text: Angle E O F , with F Image with alt text: Left parenthesis negative 5 comma negative 1 right parenthesis
Angle E O F, with FLeft parenthesis negative 2 comma negative 3 right parenthesis
Image with alt text: Angle E O F , with F Image with alt text: Left parenthesis negative 2 comma negative 3 right parenthesis
Angle E O F, with FLeft parenthesis 10 comma 2 right parenthesis
The angle that is both adjacent and supplementary to angle DOE is angle EOF, with F (-5,-1).
To determine which angles are both adjacent and supplementary to angle DOE, we need to find angles that share a side with DOE and have a combined measure of 180 degrees.
First, let's find the measure of angle DOE:
DOE = arctan((1-3)/(5-2)) = arctan(-2/3) ≈ -33.23 degrees
Now let's check each angle option:
1. Angle DOF with F(10, 2)
The measure of angle DOF = arctan((2-0)/(10-2)) = arctan(2/8) = arctan(1/4) ≈ 14.04 degrees
Since the measure of angle DOF is not equal to 180 degrees, it is not supplementary to angle DOE.
2. Angle DOF with F(-2, -3)
The measure of angle DOF = arctan((-3-0)/(-2-2)) = arctan(-3/-4) = arctan(3/4) ≈ 36.87 degrees
Since the measure of angle DOF is not equal to 180 degrees, it is not supplementary to angle DOE.
3. Angle DOF with F(-5, -1)
The measure of angle DOF = arctan((-1-0)/(-5-2)) = arctan(-1/-7) = arctan(1/7) ≈ 8.13 degrees
Since the measure of angle DOF is not equal to 180 degrees, it is not supplementary to angle DOE.
4. Angle EOF with F(-5, -1)
The measure of angle EOF = arctan((-1-0)/(-5-0)) = arctan(-1/-5) = arctan(1/5) ≈ 11.31 degrees
Since the measure of angle EOF is not equal to 180 degrees, it is not supplementary to angle DOE.
5. Angle EOF with F(-2, -3)
The measure of angle EOF = arctan((-3-0)/(-2-5)) = arctan(-3/(-7)) = arctan(3/7) ≈ 21.80 degrees
Since the measure of angle EOF is not equal to 180 degrees, it is not supplementary to angle DOE.
6. Angle EOF with F(10, 2)
The measure of angle EOF = arctan((2-0)/(10-5)) = arctan(2/5) ≈ 21.80 degrees
Since the measure of angle EOF is not equal to 180 degrees, it is not supplementary to angle DOE.
Therefore, none of the given angles are both adjacent and supplementary to angle DOE.