Angles and angle pairs are everywhere in geometry. Two types of angle pairs are complementary angles and supplementary angles.

Four types of angles in geometry are:

**Acute angles**-- measuring less than $90\xb0$ or less than $\frac{\pi}{2}$ radians**Obtuse angles**-- measuring greater than $90\xb0$ or greater than $\frac{\pi}{2}$ radians**Right angles**-- measuring exactly $90\xb0$ or exactly $\frac{\pi}{2}$ radians**Straight angles**-- measuring exactly $180\xb0$ or exactly $\pi $ radians

Complementary angles sum to exactly $90\xb0$ or exactly $\frac{\pi}{2}$ radians.

**Supplementary angles** sum to exactly $180\xb0$ or exactly $\pi $ radians.

Supplementary angles are easy to see if they are paired together, sharing a common side. Supplementary angles sharing a common side will form a straight line:

*[insert drawing of supplementary angles forming straight line]*

Supplementary angles can also share a common vertex but *not* share a common side:

Supplementary angles can also have *no* common sides or common vertex:

*[insert drawing two supplementary angles some distance apart; maybe 140° and 40°]*

Supplementary angles have two properties:

- Only two angles can sum to $180\xb0$ -- three or more angles may sum to $180\xb0$ or $\pi $ radians, but they are not considered supplementary
- The two angles must either both be
**right angles**, or one must be an**acute angle**and the other an**obtuse angle**

Here are eight sets of angles in degrees. Identify the ones that are supplementary:

- $175\xb0$ and $5\xb0$
- $137\xb0$ and $93\xb0$
- $60\xb0$ and $60\xb0$ and $60\xb0$
- $26\xb0$ and $116\xb0$
- $90\xb0$ and $90\xb0$
- $85\xb0$ and $95\xb0$
- $150\xb0$ and $31\xb0$
- $100\xb0$ and $80\xb0$

Notice the only sets that sum to $180\xb0$ are the first, fifth, sixth and eighth pairs. Only those pairs are supplementary angles. The third set has three angles that sum to $180\xb0$; three angles cannot be supplementary.

Supplementary angles are seen in three geometry theorems. Two theorems involve parallel lines.

**Congruent Supplements Theorem** -- If two angles -- we'll call them $\angle C$ and $\angle A$ -- are both supplementary to a third angle (we'll call it $\angle T$), then $\angle C$ and $\angle A$ are congruent.

We know two true statements from the theorem:

- $\angle A+\angle T=180\xb0$
- $\angle C+\angle T=180\xb0$

Since either $\angle C$ or $\angle A$ can complete the equation, then $\angle C=\angle A$.

**Same Side Interior Angles Theorem** – If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

A transversal through two lines creates eight angles, four of which can be paired off as same side interior angles.

The converse of the Same Side Interior Angles Theorem is also true. The converse theorem tells us that if a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

This is an especially useful theorem for proving lines are parallel. Here are two lines and a transversal, with the measures for two same side interior angles shown:

*[insert drawing, two parallel lines with transversal and two same side interior angles of 145° and 35° labeled with their measures]*

Since the converse of the theorem tells us the interior angles will be supplementary if the lines are parallel, and we see that $145\xb0-35\xb0=180\xb0$, then the lines must be parallel.

**Consecutive Angles in a Parallelogram** **are Supplementary** -- One property of parallelograms is that their consecutive angles (angles next to each other, sharing a side) are supplementary.

Here is parallelogram $MATH$:

*[insert drawing parallelogram MATH with ∠M = ∠T = 125° and ∠A = ∠H = 55°; label all four angles with the angle name and measure]*

Whatever angle you choose, that angle and the angle next to it (in either direction) will sum to $180\xb0$. Go ahead, try it!

This property stems directly from the Same Side Interior Angles Theorem, because any side of a parallelogram can be thought of as a transversal of two parallel sides.

A common place to find supplementary angles is in carpentry. Miter boxes, table saws and radial arm saws all depend on the user's quick mental math to find the supplementary angle to the desired angle.

Say you need a $120\xb0$ angle. You will only see numbers on those saws from $10\xb0$ to $90\xb0$. You need to know $180\xb0-120\xb0=60\xb0$, so you set the saw for a $60\xb0$ cut on the waste wood, leaving $120\xb0$ on the piece you want.

Supplementary angles also reveal themselves in repeated patterns, where right angles form windows, bricks, floor tiles, and ceiling panels. Many field stone walls have supplementary angles in them. Look around; you will see supplementary angles everywhere!

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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