First, the two triangles  and  share a vertex, so we know that  maps to  by the reflective property. Knowing this, we are able to rotate  to match the congruent sides  and .  This maps  to .  We can also note that  maps to .

Now we can reflect the triangle  across  to map  to ,  to  and  to .

So the order of rigid motions is rotation, reflection.

The definition of complementary angles is: any two angles that sum to 90.  We most often see these angles as the two angles in a right triangle that are not the right angle.  These two angles do not have to only be in right triangles, however.  Complementary triangles are any pair of angles that add up to be 90.

Even though it may not be obvious at first, the given angle is actually a supplementary angle to angle 1.  This is because the given angle is corresponding (and therefore congruent) angles to the angle adjacent to angle 1.  Since they are supplementary we can set up the following equation.

We must use the fact that lines  and  are parallel lines to solve for the missing angles.  We will break it down to solve for each angle one at a time.

Angle 1:

We know that angle 1’s supplementary angle.  Supplementary angles are two angles that add up to 180 degrees.  These two are supplementary angles because they form a straight line and straight lines are always 180 degrees.  So to solve for angle 1 we simply subtract its supplementary angle from 180.

Angle 2:

We now know that angle 1 is 130 degrees.  We can either use the fact that angles 1 and 2 are opposite vertical angles to find the value of angle 2 or we can use the fact that angle 2's supplementary angle is the given angle of 50 degrees.  If we use the latter, we would use the same procedure as last time to solve for angle 2.  If we use the fact that angles 1 and 2 are opposite vertical angles, we know that they are congruent.  Since angle  then angle .

Angle 3:

To find angle 3 we can use the fact that angles 1 and 3 are corresponding angles and therefore are congruent or we can use the fact that angles 2 and 3 are alternate interior angles and therefore are congruent.  Either method that we use will show that .

Angle 4:

To find angle 4 we can use the fact that the given angle of 50 degrees and angle 4 are alternate exterior angles and therefore are congruent, or we can use the fact that angle 3 is angle 4’s supplementary angle.  We know that the given angle and angle 4 are alternate exterior angles so .

We are able to use the relationship of opposite vertical angles to solve this problem.  The given angle and angle  are opposite vertical angles and therefore must be congruent.  So .  is supplementary to both angles  and  so they must be congruent.  and  are also opposite vertical angles so they must be congruent in that respect as well.  So

We know that lines  and  are parallel due to the information we get from the angles formed by the transversal line. 

There are:

two pairs of congruent vertically opposite angles

two pairs of congruent alternate interior angles

two pairs of congruent alternate exterior angles

two pairs of congruent corresponding angles

Just using any one of these facts is enough proof that lines  and  are parallel.

The angles formed by the transversal line intersecting lines  and  does not form congruent opposite vertical angles.  Therefore these two lines are not congruent.

To answer this question, we must understand the definition of alternate exterior angles.  When a transversal line intersects two parallel lines, 4 exterior angles are formed.  Alternate exterior angles are angles on the outsides of these two parallel lines and opposite of each other.

The figure shows a ray.  A ray has a single endpoint and the other end extends infinitely.  This is represented by an arrowhead on the end that extends infinitely.

To answer this question, we must understand the definition of vertically opposite angles.  Vertically opposite angles are angles that are formed opposite of each other when two lines intersect.  Vertically opposite angles are always congruent to each other.