Before jumping into the topic of corresponding angles, let’s first remind ourselves about angles, parallel and non-parallel lines and transversal lines.
In Geometry, an angle is composed of three parts, namely; vertex, and two arms or sides. The vertex of an angle is the point where two sides or lines of the angle meet while, arms of an angle are simply the sides of the angle.
Parallel lines are two or more lines on a 2-D plane that never meet or cross. On the other hand, non-parallel lines are two or more lines which intersect. A transversal line is a line that crosses or passes through two other lines. A transverse line can pass through two parallel or non-parallel lines.
What is a Corresponding Angle?
Angles formed when a transversal line cuts across two straight lines are known as corresponding angles. Corresponding angles are located in the same relative position an intersection of transversal and two or more straight lines.
The angle rule of corresponding angles or the corresponding angles postulate states that the corresponding angles are equal if a transversal cuts two parallel lines.
Corresponding angles are equal if the transversal line crosses at least two parallel lines.
The diagram below illustrates corresponding angles formed when a transversal line crosses two parallel lines:
From the above diagram, the pair of corresponding angles are:
< a and < e
< b and < g
< d and <f
< c and < h
Proof of Corresponding Angles
In the figure above we have two parallel lines.
We need to prove that
We have the straight angles:
From the transitive property,
From the alternate angle’s theorem,
Using substitution, we have,
Hence,
Corresponding angles formed by non-parallel lines
Corresponding angles formed when a transversal line intersects at least two non-parallel lines are not equal and in fact they don’t have any relation with each other.
Illustration:
Corresponding interior angle
A pair of corresponding angles is composed of one interior and another exterior angle. Interior angles are angles that are positioned at within the corners of the intersections.
Corresponding exterior angle
Angles that are formed outside the intersected parallel lines. An exterior angle and interior angle makes a pair of corresponding angles.
Illustration:
Interior angles include; b, c, e and f while exterior angles include; a, d, g and h.
Therefore, pairs of corresponding angles include:
< a and < e.
< b and < g
< d and < f
< c and < h
We can make the following conclusions about corresponding angles:
A pair of corresponding angles lie on the same side of the transversal.
Corresponding pair of angles comprises of one exterior angle and another interior angle.
Not all corresponding angles are equal. Corresponding angles are equal if the transversal intersects two parallel lines. If the transversal intersects non-parallel lines, the corresponding angles formed are not congruent and are not related in any way.
Corresponding angles form are supplementary angles if the transversal perpendicularly intersects two parallel lines.
Exterior angles on the same side of the transversal are supplementary if the lines are parallel. Similarly, interior angles are supplementary if the two lines are parallel.
