What makes angles parallel

This topic is relevant for:. Angles in Parallel Lines Here we will learn about angles in parallel lines including how to recognise angles in parallel lines, use angle facts to find missing angles in parallel lines, and apply angles in parallel lines facts to solve algebraic problems. Key angle facts To explore angles in parallel lines we will need to use some key angle facts. Angles in parallel lines We know that vertically opposite angles are equal and we can show this around a point within our parallel lines: If we extend the transversal line so that it crosses more parallel lines, the angles that are made are maintained throughout the diagram for any line that is parallel to the original line AB.

Alternate angles Alternate angles are angles that occur on opposite sides of the transversal line and have the same size. Other examples of alternate angles:. Corresponding angles The pairs of angles formed on the same side of the transversal that are either both obtuse or both acute and are called corresponding angles and are equal in size. Other examples of corresponding angles:. How to find a missing angle in parallel lines In order to find a missing angle in parallel lines: 1 Highlight the angle s that you already know.

Explain how to find a missing angle in parallel lines. Angles in parallel lines examples For each stage of the calculation we must clearly state any angle facts that we use. Example 5: similar triangles Show that the two triangles are similar. Common misconceptions Mixing up angle facts There are a lot of angle facts and it is easy to mistake alternate angles with corresponding angles.

Using a protractor to measure an angle. Show answer. Lines AB and CD are parallel. Learning checklist You have now learned how to: apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles understand and use the relationship between parallel lines and alternate and corresponding angles.

The next lessons are Alternate angles Corresponding angles Co-interior angles Interior angles of a polygon Exterior angles of a polygon Pythagoras theorem. We use essential and non-essential cookies to improve the experience on our website. Please read our Cookies Policy for information on how we use cookies and how to manage or change your cookie settings.

Close Privacy Overview This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. If one angle at one intersection is the same as another angle in the same position in the other intersection, then the two lines must be parallel. Two angles are corresponding if they are in matching positions in both intersections.

Alternate angles as a group subdivide into alternate interior angles and alternate exterior angles. Exterior angles lie outside the open space between the two lines suspected to be parallel. Interior angles lie within that open space between the two questioned lines. Can you identify the four interior angles? Alternate angles appear on either side of the transversal.

They cannot by definition be on the same side of the transversal. Can you find another pair of alternate exterior angles and another pair of alternate interior angles? If just one of our two pairs of alternate exterior angles are equal, then the two lines are parallel, because of the Alternate Exterior Angle Converse Theorem , which says:. Angles can be equal or congruent ; you can replace the word "equal" in both theorems with "congruent" without affecting the theorem.

You need only check one pair! Just like the exterior angles, the four interior angles have a theorem and converse of the theorem. Again, you need only check one pair of alternate interior angles! Supplementary angles create straight lines, so when the transversal cuts across a line, it leaves four supplementary angles. When a transversal cuts across lines suspected of being parallel, you might think it only creates eight supplementary angles, because you doubled the number of lines.

Those should have been obvious, but did you catch these four other supplementary angles? These four pairs are supplementary because the transversal creates identical intersections for both lines only if the lines are parallel.

The last two supplementary angles are interior angle pairs, called consecutive interior angles. If you draw a F on the diagram, the corresponding angles can be found in the corners of the F.

If two parallel lines are cut by a transversal, the corresponding angles are congruent. If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. Interior Angles on the Same Side of the Transversal: The name is a description of the "location" of the these angles. When the lines are parallel, the measures are supplementary. If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.

If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel. Vertical Angles: When straight lines intersect, vertical angles appear.

Linear Pair Angles: A linear pair are two adjacent angles forming a straight line. NOTE: The re-posting of materials in part or whole from this site to the Internet is copyright violation and is not considered "fair use" for educators. So given that, we know that not only is this side equivalent to this side, it is also equivalent to this side over here. And that tells us that that's also equivalent to that side over there. So all of these things in green are equivalent.

And by the same exact argument, this angle is going to have the same measure as this angle. And that's going to be the same as this angle, because they are opposite, or they're vertical angles. Now the important thing to realize is just what we've deduced here.

The vertical angles are equal and the corresponding angles at the same points of intersection are also equal. And so that's a new word that I'm introducing right over here. This angle and this angle are corresponding. They represent kind of the top right corner, in this example, of where we intersected. Here they represent still, I guess, the top or the top right corner of the intersection.

This would be the top left corner. They're always going to be equal, corresponding angles. And once again, really, it's, I guess, for lack of a better word, it is a bit obvious. Now on top of that, there are other words that people will see. We've essentially just proven that not only is this angle equivalent to this angle, but it's also equivalent to this angle right over here. And these two angles-- let me label them so that we can make some headway here.

So I'm going to use lowercase letters for the angles themselves. So let's call this lowercase a, lowercase b, lowercase c. So lowercase c for the angle, lowercase d, and then let me call this e, f, g, h. So we know from vertical angles that b is equal to c. But we also know that b is equal to f because they are corresponding angles.

And that f is equal to g. So vertical angles are equivalent, corresponding angles are equivalent, and so we also know, obviously, that b is equal to g. And so we say that alternate interior angles are equivalent.