Geometry Study Guide for the Math Basics

Language and Logic

Geometry was the mathematics that the ancient Greeks used. They also developed the basic foundations of arguments called logic. Because of this, most geometry courses include the basics of logic.

Postulate

A postulate is a statement which is taken to be true without proof. There are five main postulates of (Euclidean) geometry.

1) A straight line segment can be drawn between any two points.

2) Any straight line segment can be extended indefinitely to create a line.

3) Given a line segment, a circle can be drawn having the segment as its radius and one endpoint as its center.

4) All right angles are congruent.

5) Parallel postulate: Given a line and a point on the line, only one line can be drawn through that point which is parallel to the first line.

Postulates are considered the foundations of any argument. You’ll encounter a variety of other postulates throughout your geometry course.

Theorem

A theorem is a statement which can be proven true based on postulates, definitions, and/or previously proven theorems.

You’ll learn many theorems throughout your geometry courses. We suggest keeping a list of them in a special place as you learn them.

Some common theorems: The sum of angles of a triangle is \(180^\circ\). The intersection of two planes is a line. If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Proof

Math is inherently true because it can be proven using the language of logic. Each step must be justified by a postulate, definition, theorem, or property. There are a few different ways to prove a statement.

Algebraic

If you want to be very thorough when solving an equation, you can justify each step by saying the property or identity used for each step.

Equation: \(2x+5=13\)

Step 1) Use the subtraction property of equality to subtract 5 from each side of the equation.

\[2x +5 -5 = 13 -5\] \[2x + 0 = 8\]

Step 2) Since 0 is the additive identity,

\[2x + 0 = 8\] \[2x = 8\]

Step 3) Now, use the division property of equality and divide both sides of the equation by 2.

\[\dfrac{2x}{2} = \dfrac{8}{2}\] \[1x = 4\]

Step 4) Since 1 is the multiplicative identity, \(1x=4 \rightarrow x=4\), which is the solution.

Two Column

A two column proof is a nice way to organize each step you take paired with the reason you are taking it. A geometric proof is generally initiated by a true statement called the given. It’s always the step and reason of a two column proof. Here’s the two column proof of the same algebra problem.

\[\begin{array}{|c|c|} \hline \textbf{Statement} & \textbf{Reason} \\ \hline 2x+5=13 & \text{Given} \\ \hline 2x +0 =8 & \text{Subtraction property of equality} \\ \hline 2x = 8 & \text{Additive identity} \\ \hline 1x=4 & \text{Division property of equality} \\ \hline x=4 & \text{Multiplicative identity} \\ \hline \end{array}\]

Here’s a two column proof you might see in geometry.

Given: \(\overline{AD} \perp \overline{BC}\), and \(\angle{B} \cong \angle{C}\).

Prove: \(\triangle{ADB} \cong \triangle{ADC}\)

\[\begin{array}{|c|c|} \hline \textbf{Statement} & \textbf{Reason} \\ \hline \overline{AD} \perp \overline{BC}, \;and\; \angle{B} \cong \angle{C} & \text{Given} \\ \hline \angle{ADB} \text{ and } \angle{ADC} \text{ are right angles} & \text{definition of }\perp \\ \hline \angle{ADB} \cong \angle{ADC} & \text{All right angles are congruent} \\ \hline \overline{AD} \cong \overline{AD} & \text{reflexive property of congruence} \\ \hline \triangle{ADB} \cong \triangle{ADC} & \text{AAS} \\ \hline \end{array}\]

Paragraph

If you’re fairly comfortable with organizing your thoughts, perhaps you prefer paragraph proofs. Here’s the same proof as above in paragraph format.

We are given that \(\overline{AD} \perp \overline{BC}\), and \(\angle{B} \cong \angle{C}\). First, infer that \(\angle{ADB}\) and \(\angle{ADC}\) are right angles by the definition of perpendicularity. Furthermore, because all right angles are congruent, \(\angle{ADB} \cong \angle{ADC}\). It almost goes without saying that \(\overline{AD} \cong \overline{AD}\) by the reflexive property of congruence. Finally, by the AAS congruence theorem, we can say that \(\triangle{ADB} \cong \triangle{ADC}\).

If-Then Statements and Truth Values

Any postulate, theorem, and definition you learn in your geometry course is an example of an If-Then statement. Each of these statements has two parts: the If part is the hypothesis, and the Then part is the conclusion.

Conditional Any statement of the form If P, then Q is called a conditional statement. For example, If a rhombus has a right angle, then it is a square, is a conditional.

Converse

The converse of a conditional (If P, then Q) is the statement If Q, then P. Take the conditional: If an angle is acute, then its measure is less than \(90^\circ\). The converse is: If the measure of an angle is less than \(90^\circ\), then it is acute.

Inverse

The inverse of a conditional (If P, then Q), is the statement If not P, then not Q. Simply, you negate both the hypotheses and the conclusion of the original conditional. Take the following conditional: If it is raining, then the grass is wet. The inverse would be: If it is not raining, then the grass is not wet.

Contrapositive

Finally, the contrapositive can be described as the inverse converse of the conditional. So, the contrapositive of If P, then Q would be If not Q, then not P. So, the contrapositive of If you tell a funny joke, then I will laugh is If I don’t laugh, then you didn’t tell a funny joke.

Truth Values and Biconditionals

There are two options for the truth value of a conditional: true or false. Conditional statements are only true if the conclusion follows directly as a result of the hypothesis. Otherwise, it is false (even if the conclusion and hypothesis have nothing to do with another).

Let’s look at the following example and identify the truth value of each statement. Conditional: If a figure is a square, then it is a parallelogram. TRUE

Converse: If a figure is a parallelogram, then it is a square. FALSE (it could be a rectangle)

Inverse: If a figure is not a square, then it is not a parallelogram. FALSE (it could be a rhombus, which is a parallelogram)

Contrapositive If a figure is not a parallelogram, then it is not a square. TRUE

Notice the conditional and the contrapositive have the same truth values, as do the converse and the inverse. This is always the case.

Occasionally, the conditional and the converse are both true (and thus, so are the inverse and the contrapositive). If this happens, a new statement can be formed called a biconditional: P if and only if Q. Look at the following conditional and converse, followed by the resulting biconditional:

Conditional: If a polygon is a quadrilateral, then it has 4 sides. TRUE
Converse: If a polygon has 4 sides, then it is a quadrilateral. TRUE
Biconditional: A polygon is a quadrilateral if and only if it has 4 sides.

Every definition is always a biconditional.

The Distance Formula

You can’t go through geometry without being very familiar with the distance formula. This formula finds the distance between any two points on a coordinate plane. Given two points: \(A = (x_1,y_1)\) and \(B=(x_2, y_2)\), then \(D(A,B)=\sqrt{(x_1 - x_2)^2 + (y_1 -y_2)^2}\)

Find the distance between \((1,-3)\) and \((4, 1)\).

\[D = \sqrt{(1-4)^2 + (-3-1)^2}\\ D = \sqrt{(-3)^2 + (-4)^2}\\ D = \sqrt{9 + 16}\\ D = \sqrt{25}\\ D = 5\]