Supplementary angles are any two angles that sum (add) up to 180° (a straight line).

Complementary angles are any two angles that sum (add) up to 90° (a corner).

In the above image, we see that angles a and b sum up to 90°, therefore they are complementary angles.

In the above image, we see that angles c and d sum up to 180°, therefore they are supplementary angles.

While the angles were drawn adjacent to one another, they don’t have to be – although, that is the most common way you will encounter them – any two angles that add up to 90° or 180° are complementary or supplementary angles.

Why is this Useful?

This is useful, because it helps in solving trigonometric problems.

For example, consider a construction in which A and B are supposed to be perpendicular. Unfortunately, you can’t measure the angle directly because there is a support beam that comes out from the corner. How can you determine if the walls are perpendicular?

We know that walls (or beams or forces or whatever) are perpendicular if the angle between them is 90°

We can measure angle α and angle β and if they are complimentary, then we know the walls are perpendicular.

The post Understanding Supplementary and Complementary Angles appeared first on Complete, Concrete, Concise.

This is illustrated in the image below:

We see two parallel lines and a third line (transversal) intersecting (crossing or cutting through) both of them. The green shaded angles are: (1) inside (between) the two parallel lines, (2) congruent (identical or the same), and (3) on opposite sides of the transversal.

This is true for the other two unshaded interior angles. It is also true for the alternate exterior angles (but not proved here).

Axioms

Proofs are built on two things: (1) postulates, axioms, or hypotheses – these are things that are assumed to be true, but can’t be proved. In general, we try to use the fewest number of axioms we can; (2) other proofs.

This proof depends on two axioms: (1) if you pick any two distinct points on a straight line, the angle between those two points will be 180°; (2) if you take any two intersecting straight lines and shift one of the lines so it it is in a different position, but still parallel to its original position, the angle between the two intersecting lines stays the same.

Proof

1) Consider two intersecting lines:

At the point of intersection, there is an angle (which I call A).

2) If the line is straight, the angle between any two points on that line must be 180° (axiom #1 from above):

3) If one angle is A degrees, then obviously the other angle must be 180° – A:

4) This is true for both straight lines:

5) This means the remaining angle must be 180° – (180° – A) = A degrees:

This proves that angles on alternate sides of the transversal (at the point of intersection) are congruent (identical).

6) If we draw a line parallel to one of the lines (it doesn’t matter which) and it intersects the other line (that line is now called a transversal), we know that the angles of intersection must be the same (axiom #2 from above).

If the angles of intersection are the same for both lines, then the alternate interior angles must be the same:

The post Proving Alternate Interior Angles are Congruent (the same) appeared first on Complete, Concrete, Concise.