one of the exterior angles of a triangle is 126 degree and the interior opposite angles are in the ratio 2:4. Find the angles of the triangle.
"An measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles."
http://www.regentsprep.org/Regents/math/geometry/GP5/LExtAng.htm
126/6 = 21
42 and 84
To find the angles of the triangle, we need to use the properties of triangles and solve for the missing angles. Let's go step-by-step:
Step 1: Determine the ratio of the interior opposite angles.
The ratio given is 2:4. Since the sum of the interior opposite angles of a triangle is always 180 degrees, we can express this ratio as:
2x + 4x = 180
6x = 180
x = 30
Step 2: Calculate the interior opposite angles.
Now that we have the value of x, we can substitute it into the ratio to find the measurements of the interior opposite angles:
2x = 2(30) = 60 degrees
4x = 4(30) = 120 degrees
Step 3: Calculate the third interior angle.
Since the sum of all three interior angles of a triangle is 180 degrees, we can subtract the known angles from 180 to find the third interior angle:
Angle3 = 180 - (60 + 120)
Angle3 = 180 - 180
Angle3 = 0 degrees
Step 4: Calculate the exterior angle.
The exterior angle is equal to the sum of the opposite interior angles. In this case, it is given as 126 degrees:
Exterior angle = 60 +120
Exterior angle = 180 degrees
Step 5: Calculate the remaining exterior angles.
Since the sum of all exterior angles of a triangle is always 360 degrees, we can subtract the known exterior angle from 360 to find the remaining exterior angles:
Remaining exterior angles = 360 - 180
Remaining exterior angles = 180 degrees
Step 6: Determine the angles of the triangle.
Now, we have the measurements for all three interior angles of the triangle:
Angle1 = 60 degrees
Angle2 = 120 degrees
Angle3 = 0 degrees
And the measurements for all three exterior angles of the triangle:
Exterior angle1 = 126 degrees
Exterior angle2 = 180 degrees
Exterior angle3 = 180 degrees
Please note that the value of Angle3 is 0 degrees, which means that this "triangle" does not exist.
To find the angles of the triangle, let's first determine the measures of the interior angles.
We know that one of the exterior angles of the triangle is 126 degrees. The sum of the measures of the exterior angles of any polygon is always 360 degrees. Therefore, we can find the measure of the other two exterior angles by subtracting the given exterior angle from 360 degrees:
Other exterior angle = 360 degrees - 126 degrees = 234 degrees
Now, we have the measures of the exterior angles. Let's use this information to find the measures of the interior angles.
The exterior angle of a triangle is equal to the sum of the two corresponding interior opposite angles. Therefore, we can write the following equation:
Exterior angle = Interior opposite angle 1 + Interior opposite angle 2
Substituting the given values, we get:
126 degrees = Interior opposite angle 1 + Interior opposite angle 2
We are also given that the interior opposite angles are in a ratio of 2:4. We can represent the interior opposite angles as 2x and 4x, where x is a constant.
Plugging in these values, we have:
126 degrees = 2x + 4x
Combining like terms:
126 degrees = 6x
To solve for x, divide both sides of the equation by 6:
x = 126 degrees / 6 = 21 degrees
Now that we have the value of x, we can find the measures of the interior angles:
Interior opposite angle 1 = 2x = 2 * 21 = 42 degrees
Interior opposite angle 2 = 4x = 4 * 21 = 84 degrees
Therefore, the three angles of the triangle are:
Angle 1 = Interior opposite angle 1 = 42 degrees
Angle 2 = Interior opposite angle 2 = 84 degrees
Angle 3 = Other exterior angle = 234 degrees