How can we use the concept:
Once we have identified that we have alternate interior angles, we know that they are congruent. This allows us to solve for missing angles! For example, given that the following lines are parallel, we know that angle x is 40° because these are alternate interior angles and, therefore, they are congruent!
Once we have identified that we have alternate interior angles, we know that they are congruent. This allows us to solve for missing angles! For example, given that the following lines are parallel, we know that angle x is 40° because these are alternate interior angles and, therefore, they are congruent!
Sample Math Problems
Question
Given that l||m, what is the measure of angle r?
Solution
We can see that angle r and the 98° angle are alternate interior angles and therefore must be congruent. So the measure of angle r is 98°.
Question
Given that p || r, solve for x.
Solution
Since these angles are alternate interior angles, they must be congruent (in other words, equal to each other). So 2x+34=52. When I solve this equation, I find that x=9.
Question
In the following diagram, r || p || m and m∠4=117°. Name all the angles equal to 63°
Solution
If m∠4=117°, then m∠1=63° because ∠4 and ∠1 form a line together, which equals 180°. Therefore m∠2=63° because ∠2 and ∠1 are vertical angles (across from each other on an intersection) and therefore must be congruent. ∠2 and ∠8 are alternate interior angles, so they must be congruent. ∠6 and ∠8 are another pair of vertical angles that are congruent. And last but not least, ∠6 and ∠9 are another pair of alternate interior angles and, therefore, congruent. I cannot draw any conclusions about angles 10 through 18 because they are on a different transversal than the angle I was given. So angles 1, 2, 8, 6, and 9 are equal to 63°.
Question
In the figure below, l || m and x=50. What is z-y?
Solution
If x=50, then y=50 because they are alternate interior angles. Therefore, z is 130 because y and z form a line together, which is 180°. So z-y = 130 - 50 = 80.
