When you learn calculus in class or study it from a textbook it can be very overwhelming. The way the concept is explained in class or in a book can sometimes take a while to understand and you may find that after hours of research, you are no wiser than when you first began. Maths can be difficult, but don’t worry!
If you’re struggling with understanding alternate interior angles, this article is for you. The following will help you understand what these angles are and explore examples, and frequently asked questions.
What are Alternate Interior Angles?
Put simply, alternate interior angles are formed when two lines are intersected by a third line. That third line is known as the transversal line.
If two parallel or non-parallel lines are intersected by a transversal line, alternate interior angles are always formed. The angles are on opposite sides of the transversal line and on the inside of the parallel or non-parallel lines.
This is why they are called alternate interior angles – they are opposite, and on the interior of the two lines! Note that you will find two pairs of alternate interior angles on the transversal line.
If you consider the two angles on the same side as the transversal line, these are known as consecutive interior angles.
If the lines intersected by the transversal are parallel, alternate interior angles are equal. Likewise, if you knew that the alternate interior angles were equal, then you could confidently say the two lines were parallel.
Properties of Alternate Interior Angles
To help you understand more clearly what alternate interior angles are, consider the following properties:
- Alternate Interior angles that are opposite each other are congruent (the same measure).
- Alternate Interior angles that are on the same side of the transversal line are called consecutive interior angles.
- In the case of non-parallel sides, alternate interior angles have no special properties and are not congruent.
Alternate Interior Angles Theorem
A theorem is a hypothesis that can be proven true by using maths. A proof is the method of demonstrating the validity of a theorem. If two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent, according to the Alternate Interior Angles Theorem.
Examples of Alternate Interior Angles
We have two parallel lines, and our task here is to prove that Y= 122° by using the theorem explained above.
- According to corresponding angles, angle X is equal to angle K.
- Angles on a straight line equal 180 °. This means 122° + Angle X = 180°. This is the same for angles K and Y.
- As angle X is the same as angle K, it means that if you subtract angle K (58°) from 180°, it would give you angle Y. This is shown below:
180° – 122°= Angle X (58°).
And Angle X is the same as Angle K
180° – 58° (Angle K) = Angle Y (122°)
4. Therefore, both highlighted angles are 122°, meaning the alternate interior angles are congruent.
Frequently Asked Questions
Can alternate interior angles be 90°?
Though very uncommon. If two parallel lines are joined by a line perpendicular to them, the resulting alternate interior angles will be 90 degrees. Understand, however, that this is very unlikely to be seen in class or in an exam.
Do alternate interior angles add up to 180°?
Angles which add up to 180° are called supplementary angles. We know that adjacent angles on a straight line always add up to 180° but it’s also true that interior angles add up to 180°.
What about alternate interior angles? Unless the alternate interior vertical angles are 90° then they will not add up to 180°. If the alternate interior angles are obtuse, then adding them together will result in a number higher than 180°. Therefore, if the alternate interior angles are acute, then adding them together will result in a number below 180°.
Still having trouble?
If you’re still struggling with alternate interior angles or anything related to math, you may want to consider hiring a tutor to gain a better understanding of the topic and get personalized support.
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