# Angle Properties, Postulates, and Theorems

In order to study geometry

in a logical way, it will be important to understand key mathematical properties

and to know how to apply useful postulates and theorems. A **postulate** is a

proposition that has not been proven true, but is considered to be true on the basis

for mathematical reasoning. **Theorems**, on the other hand, are statements that

have been proven to be true with the use of other theorems or statements. While

some postulates and theorems have been introduced in the previous sections, others

are new to our study of geometry. We will apply these properties, postulates, and

theorems to help drive our mathematical proofs in a very logical, reason-based way.

Before we begin, we must introduce the concept of congruency. Angles are **congruent**

if their measures, in degrees, are equal. ** Note**: “congruent” does not

mean “equal.” While they seem quite similar, congruent angles do not have to point

in the same direction. The only way to get equal angles is by piling two angles

of equal measure on top of each other.

## Properties

We will utilize the following properties to help us reason through several geometric

proofs.

### Reflexive Property

A quantity is equal to itself.

### Symmetric Property

If * A = B*, then

*.*

**B = A**### Transitive Property

If * A = B* and

*, then*

**B = C***.*

**A = C**### Addition Property of Equality

If * A = B*, then

*.*

**A + C = B + C**## Angle Postulates

### Angle Addition Postulate

If a point lies on the interior of an angle, that angle is the sum of two smaller

angles with legs that go through the given point.

Consider the figure below in which point * T* lies on the interior of

**. By this postulate, we have that**

*?QRS**.*

**?QRS = ?QRT + ?TRS**We have actually applied this postulate when we practiced finding the complements

and supplements of angles in the previous section.

### Corresponding Angles Postulate

If a transversal intersects two **parallel** lines, the pairs of corresponding

angles are congruent.

__Converse also true__: If a transversal intersects two lines and the corresponding

angles are congruent, then the lines are parallel.

*The figure above yields four pairs of corresponding angles.*

### Parallel Postulate

Given a line and a point not on that line, there exists a unique line through the

point parallel to the given line.

The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry.

*There are an infinite number of lines that pass through point E, but only
the red line runs parallel to line CD. Any other line through E will
eventually intersect line CD.*

## Angle Theorems

### Alternate Exterior Angles Theorem

If a transversal intersects two **parallel** lines, then the alternate exterior

angles are congruent.

__Converse also true__: If a transversal intersects two lines and the alternate

exterior angles are congruent, then the lines are parallel.

*The alternate exterior angles have the same degree measures because the lines are
parallel to each other.*

### Alternate Interior Angles Theorem

If a transversal intersects two **parallel** lines, then the alternate interior

angles are congruent.

__Converse also true__: If a transversal intersects two lines and the alternate

interior angles are congruent, then the lines are parallel.

*The alternate interior angles have the same degree measures because the lines are
parallel to each other.*

### Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the

two angles are congruent.

### Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the

two angles are congruent.

### Right Angles Theorem

All right angles are congruent.

### 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.

__Converse also true__: If a transversal intersects two lines and the interior

angles on the same side of the transversal are supplementary, then the lines are

parallel.

*The sum of the degree measures of the same-side interior angles is 180°.*

### Vertical Angles Theorem

If two angles are vertical angles, then they have equal measures.

*The vertical angles have equal degree measures. There are two pairs of vertical angles.*

## Exercises

**(1) Given: m?DGH = 131**

**Find: m?GHK**

First, we must rely on the information we are given to begin our proof. In this

exercise, we note that the measure of ** ?DGH** is

*.*

**131°**
From the illustration provided, we also see that lines ** DJ** and

*EK*are parallel to each other. Therefore, we can utilize some of the angle theorems

above in order to find the measure of

**.**

*?GHK*
We realize that there exists a relationship between ** ?DGH** and

**:**

*?EHI*they are corresponding angles. Thus, we can utilize the

**Corresponding Angles Postulate**

to determine that

*.*

**?DGH??EHI**
Directly opposite from ** ?EHI** is

**. Since they are**

*?GHK*vertical angles, we can use the

**Vertical Angles Theorem**, to see that

**.**

*?EHI??GHK*
Now, by **transitivity**, we have that ** ?DGH??GHK**.

**Congruent angles** have equal degree measures, so the measure of **?DGH**

is equal to the measure of ** ?GHK**.

Finally, we use **substitution** to conclude that the measure of **?GHK**

is ** 131°**. This argument is organized in two-column proof form below.

**(2) Given: m?1 = m?3**

**Prove: m?PTR = m?STQ**

We begin our proof with the fact that the measures of ** ?1** and

*?3*are equal.

In our second step, we use the **Reflexive Property** to show that *?2*

is equal to itself.

Though trivial, the previous step was necessary because it set us up to use the

**Addition Property of Equality** by showing that adding the measure of *?2*

to two equal angles preserves equality.

Then, by the **Angle Addition Postulate** we see that ** ?PTR** is the

sum of

**and**

*?1***, whereas**

*?2***is the**

*?STQ*sum of

**and**

*?3***.**

*?2*
Ultimately, through **substitution**, it is clear that the measures of *?PTR*

and ** ?STQ** are equal. The two-column proof for this exercise is shown

below.

**(3) Given: m?DCJ = 71, m?GFJ = 46**

**Prove: m?AJH = 117**

We are given the measure of ** ?DCJ** and

**to begin the**

*?GFJ*exercise. Also, notice that the three lines that run horizontally in the illustration

are parallel to each other. The diagram also shows us that the final steps of our

proof may require us to add up the two angles that compose

**.**

*?AJH*
We find that there exists a relationship between ** ?DCJ** and

**:**

*?AJI*they are alternate interior angles. Thus, we can use the

**Alternate Interior Angles**

Theoremto claim that they are congruent to each other.

Theorem

By the definition of **congruence**, their angles have the same measures, so

they are equal.

Now, we **substitute** the measure of ** ?DCJ** with

*71*since we were given that quantity. This tells us that

**is also**

*?AJI***.**

*71°*
Since ** ?GFJ** and

**are also alternate interior angles,**

*?HJI*we claim congruence between them by the

**Alternate Interior Angles Theorem**.

The definition of congruent angles once again proves that the angles have equal

measures. Since we knew the measure of ** ?GFJ**, we just

**substitute**

to show that

**is the degree measure of**

*46***.**

*?HJI*
As predicted above, we can use the **Angle Addition Postulate** to get the sum

of ** ?AJI** and

**since they compose**

*?HJI***.**

*?AJH*Ultimately, we see that the sum of these two angles gives us

**.**

*117°*The two-column proof for this exercise is shown below.

**(4) Given: m?1 = 4x + 9, m?2 = 7(x + 4)**

**Find: m?3**

In this exercise, we are not given specific degree measures for the angles shown.

Rather, we must use some algebra

to help us determine the measure of ** ?3**. As always, we begin with the

information given in the problem. In this case, we are given equations for the measures

of

**and**

*?1***. Also, we note that there exists two pairs**

*?2*of parallel lines in the diagram.

By the **Same-Side Interior Angles Theorem**, we know that that sum of *?1*

and ** ?2** is

**because they are supplementary.**

*180*
After **substituting** these angles by the measures given to us and simplifying,

we have ** 11x + 37 = 180**. In order to solve for

*, we*

**x**first subtract both sides of the equation by

**, and then divide both sides by**

*37***.**

*11*
Once we have determined that the value of * x* is

**, we plug it back in to the equation for the measure**

*13*of

**with the intention of eventually using the**

*?2***Corresponding Angles**

Postulate. Plugging

Postulate

**in for**

*13***gives us a measure of**

*x***for**

*119***.**

*?2*
Finally, we conclude that *?3*

must have this degree measure as well since ** ?2** and

*?3*are

**congruent**. The two-column proof that shows this argument is shown below.