# What is alternate interior angles?

Let be a set of lines in the plane. A line *k* is
**transversal** of if # , and # for all . Let be transversal to
*m* and *n* at points *A* and *B*,
respectively. We say that each of the angles of intersection of and
*m* and of and *n* has a *transversal side* in and a
*non-transversal side* not contained in . **Definition:**
An angle of intersection of *m* and *k* and one of
*n* and *k* are **alternate interior angles** if their
transversal sides are opposite directed and intersecting, and if
their non-transversal sides lie on opposite sides of . Two of these
angles are **corresponding angles** if their transversal sides
have like directions and their non-transversal sides lie on the
same side of . **Definition:** If *k* and are lines so that
, we shall call these lines **non-intersecting**. We want to
reserve the word parallel for later. **Theorem 9.1:**[Alternate
Interior Angle Theorem] *If two lines cut by a transversal have a
pair of congruent alternate interior angles, then the two lines are
non-intersecting.*

**Figure 10.1:** Alternate interior angles

**Proof:** Let *m* and *n* be two lines cut by the
transversal . Let the points of intersection be *B* and
*B*', respectively. Choose a point *A* on *m* on one
side of , and choose on the same side of as *A*. Likewise,
choose on the opposite side of from *A*. Choose on the same
side of as *C*. Hence, it is on the opposite side of from
*A*', by the *Plane Separation Axiom*. We are given that
. Assume that the lines *m* and *n* are not
non-intersecting; *i.e.*, they have a nonempty intersection.
Let us denote this point of intersection by *D*. *D* is
on one side of , so by changing the labeling, if necessary, we may
assume that *D* lies on the same side of as *C* and
*C*'. By *Congruence Axiom 1* there is a unique point so
that . Since, (by Axiom C-2), we may apply the SAS Axiom to prove
that

From the definition of congruent triangles, it follows that .
Now, the supplement of is congruent to the supplement of , by
Proposition 8.5. The supplement of is and . Therefore, is congruent
to the supplement of . Since the angles share a side, they are
themselves supplementary. Thus, and we have shown that or that is
more that one point, contradicting Proposition 6.1. Thus, *m*
and *n* must be non-intersecting. **Corollary 1:** *If m
and n are distinct lines both perpendicular to the line , then m
and n are non-intersecting.* **Proof:** is the transversal to
*m* and *n*. The alternate interior angles are right
angles. By Proposition 8.14 all right angles are congruent, so the
*Alternate Interior Angle Theorem* applies. *m* and
*n* are non-intersecting. **Corollary 2:** *If P is a
point not on , then the perpendicular dropped from P to is
unique.* **Proof:** Assume that *m* is a perpendicular
to through *P*, intersecting at *Q*. If *n* is
another perpendicular to through *P* intersecting at *R*,
then *m* and *n* are two distinct lines perpendicular to
. By the above corollary, they are non-intersecting, but each
contains *P*. Thus, the second line cannot be distinct, and
the perpendicular is unique. The point at which this perpendicular
intersects the line , is called the *foot* of the
perpendicular

### What is the difference between alternate interior angles and interior angles?

An interior angle is an angle defined by two sides of a polygon and that is inside the polygon. Opposite interior angles are specific pairs of interior angles, those that are opposite each other in the polygon. Alternate or opposite interior angles are also angles that lie on opposte sides of the tranversal line that cuts through parallel lines.

### Which pair of angles are alternate interior angles?

_\_________ .a\b _c\d________ .....\ When a line crosses 2 lines, 8 angles are formed. Four are exterior angles - outside the 2 lines, and four are interior angles. These are labelled a, b, c, d in the diagram. a & d are alternate interior angles because they alternate from one side of the intersecting line to the other; b & c are also alternate interior angles. They are also known as "Z-angles" because the top…

### What are the conditions that guarantee parallelism?

Parallel lines are lines that are coplanar (lying on the same plane) and do not intersect when cut by a transversal, corresponding angles formed by line n are equal in measure, alternate interior angles are equal in measures, the measures of alternate exterior angles are equal, consecutive interior angles are supplementary, consecutive exterior angles are supplementary.

### Uses of parallel lines cut by a transversal?

There's lots of useful things you can discover when parallel lines are cut by a transversal, most of them having to do with angle relationships. Corresponding angles are congruent, alternate interior angles are congruent, same side or consecutive interior angles are supplementary, alternate exterior angles are congruent, and vertical angles are congruent.