Practice and master geometry concepts with helpful walkthroughs from our elite math educators, then try practice math problems to sharpen your skills.
By Ellen Rattin
Alternate Interior Angles are pairs of angles formed when parallel lines are crossed by a transversal. They are defined as being inside the parallel lines (interior) and on opposite sides of the transversal (alternate). A very important property of alternate interior angles is that they are congruent.
Why is this concept useful?
Where does this concept fit into the curriculum?
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!
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.
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Get PDF1. Given that l||m, what is the measure of angle r?
2. Given that p || r, solve for x.
3. In the following diagram, r || p || m and m∠12=30°. Name all the angles equal to 30°.
4. In the figure below, l || m. Which of the given answer choices must equal 180°? Select all that apply.
A. ∠1 + ∠3 + ∠4
B. ∠5 + ∠3 + ∠4
C. ∠1 + ∠3 + ∠2
D. ∠2 + ∠3 + ∠5
E. ∠1 + ∠2 + ∠4