The sum of the measures of two alternate interior angles formed by two parallel lines with a transversal is 210 degrees. What are the possible measures of the angles formed by the parallel lines and the transversal?
Since alternate interior angles are congruent, let's call the measures of the two angles $x$ and $x$.
The sum of these two angles is $x+x=2x=210^\circ$. Therefore, we have $2x=210^\circ$.
Dividing both sides by $2$, we find $x=\frac{210^\circ}{2}=105^\circ$.
Therefore, the measures of the angles formed by the parallel lines and the transversal are $\boxed{105^\circ}$ for both angles.
To find the measures of the angles formed by the parallel lines and the transversal, we need to understand the properties of alternate interior angles.
Alternate interior angles are formed when a transversal intersects two parallel lines. These angles are located on opposite sides of the transversal and are congruent (meaning they have the same measure).
Let's denote the measures of the two alternate interior angles as x and y. Since the sum of the measures of the two angles is 210 degrees, we can write the equation:
x + y = 210
Now, let's solve for the possible measures of the angles:
1. Let's start by assuming x has the smallest possible value, which is 0 degrees. In this case, y would have to be 210 degrees for the sum to equal 210 degrees.
Therefore, one possible solution is x = 0 degrees and y = 210 degrees.
2. Now, let's assume x has a larger value. Let's say x = 30 degrees. In this case, y would have to be 180 degrees to make the sum equal 210 degrees.
Therefore, another possible solution is x = 30 degrees and y = 180 degrees.
3. We can continue this process and assume larger values for x. For example, if we let x = 60 degrees, then y would be 150 degrees.
Thus, another possible solution is x = 60 degrees and y = 150 degrees.
Overall, the possible measures of the angles formed by the parallel lines and the transversal are:
x = 0 degrees, y = 210 degrees
x = 30 degrees, y = 180 degrees
x = 60 degrees, y = 150 degrees
These are the solutions given the information provided.
To find the measures of the angles formed by the parallel lines and the transversal, we need to understand alternate interior angles and their properties.
When two parallel lines are intersected by a transversal, alternate interior angles are formed on opposite sides of the transversal and between the parallel lines. These angles have equal measures.
Let's denote one of the alternate interior angles as 'x'. Since the sum of the measures of these angles is 210 degrees, we can set up an equation:
x + x = 210
Combining like terms, we get:
2x = 210
To solve for 'x', divide both sides of the equation by 2:
x = 210/2
x = 105
Therefore, the measure of one alternate interior angle is 105 degrees. Since alternate interior angles have equal measures, the other angle will also be 105 degrees.