*Contributor: Marlene Vogel. Lesson ID: 11284*

Does becoming a geometry scholar sound like an impossible mission? Not when Ms. Mars is the teacher! Follow step-by-illustrated-step to learn all the angles of geometric theory, and solve the mission!

categories

subject

Math

learning style

Visual

personality style

Golden Retriever

Grade Level

High School (9-12)

Lesson Type

Dig Deeper

Welcome to Team Superhero! Your mission, should you choose to accept it, is to use your powers of deductive reasoning to match the clues with the right person. We have given you a document called *The Mission Is Possible* to help you complete your mission. Good luck! This message will not self-destruct in 5 seconds. (Complete this activity, found in the **Downloadable Resources** in the right-hand sidebar, after the lesson.)

Before you begin your mission, here are a few terms you need to know to be successful when using deductive reasoning:

The definitions can be found at Mathwords.com. Write them down and get to know them!

*deductive reasoning**vertical angles**adjacent angles**complementary angles**supplementary angles**vertex**acute angles**theorem*

This lesson will help you learn what *deductive reasoning* is and how to use it, and will introduce you to some of the *theorems* used in geometry.

One helpful way of learning a new math concept is to review a previously learned math concept that is similar to the new one.

Geometry has three rules of *congruency* (geometric figures being equal). A great way to understand these three rules is to review the Properties of Equality and Real Numbers from Algebra.

**Properties of Equality and Real Numbers**

**Addition Property**

This algebraic property states that if one number is equal to another number, then each number added to a third number is equal as well.

For example, if 4 + 3 = 6 + 1, then 4 + 3 + 2 = 6 + 1 + 2.

In this example, the expressions in the first part each equal 7. In the second part of the example, each expression is added to two. As you can see, their sums are still equal.

*If a = b, then a + c = b + c.*

**Subtraction Property**

This algebraic property states that if one number is equal to another number and each number is subtracted by the same number, they will still be equal.

For example, if 4 + 3 = 6 + 1, then 4 + 3 – 2 = 6 + 1 – 2.

As you can see, the two expressions in the beginning of the example equal 7, and the expressions in the second half of the example are subtracted by the number 2, which gives each expression the sum of 5.

*If a = b, then a - c = b - c*

**Multiplication Property**

This algebraic property states that if one number is equal to another and each number is multiplied by the same number, they will still be equal.

For example, if 4 + 3 = 6 + 1, then (4 + 3) x 2 = (6 + 1) x 2.

As you can see, the expressions in the beginning of the example both equal 7. Once each expression is multiplied by 2, they are still equal to each other; both having a product of 14.

*If a = b, then (a)(c) = (b)(c)*

**Division Property**

This algebraic property states that if one number is equal to another and each number is divided by the same number, they will still be equal.

For example, if 5 + 5 = 8 + 2, then (5 + 5) ÷ 2 = (8 + 2) ÷ 2 is equal as well.

As you can see in the example, the expressions in the beginning of the example both have an answer of 10; therefore, they are equal. The second half of the equation shows that each expression divided by 2 will result in the same quotient, 5. Therefore, both of the expressions in the second half of the example are equal.

*If a = b, then a ÷ c = b ÷ c*

**Substitution Property**

This algebraic property states that if a = b, then b can be substituted for a in any equation.

**Distributive Property**

This algebraic property states that if you are multiplying an expression by a number, you can multiply each number in the expression by that number, add the products, and arrive at the same answer.

For example, if you add 4 and 5 and multiply the sum by 2, the product is 18. In comparison, if you multiply 4 and 2 to obtain the product 8 and you multiply 5 and 2 to obtain the product 10, then add the sums 8 and 10, you arrive at the sum 18, which is the same answer as the first part of the example.

*a(b = c) = ab + ac*

Now that you have reviewed the algebraic properties, it may be easier for you to understand the Properties of Congruence from geometry. An important point to note is that the Properties of Congruence were developed from the Properties of Equality.

**Reflexive Property**

Any geometric figure is congruent to itself. Remember, *congruent* means the two geometric figures have the same size and shape. This property can be used with line segments, angles, and geometric shapes. The symbol used to show that two geometric figures are congruent is ≅.

The way to show that two *line segments* are congruent to each other is below:

The way to show that two *angles* are congruent to each other is below:

The way to show that two *triangles* are congruent to each other is below:

A great way to remember the reflexive property is to think of the word "reflection," which is similar to "reflexive." As you know, a reflection is a copy of the original object. This geometric property is similar to the addition, subtraction, multiplication, and division properties in algebra.

**Symmetric Property**

This geometric property states that each side of an equation can be swapped with the other side.

Below is an example of how to use the symmetric property with *line segments*. Read the statement aloud. It states that if line segment AB is congruent to line segment CD, then line segment CD is congruent to line segment AB.

Below is an example of how to use the symmetric property with *angles*. Read the statement aloud. It states that if angle A is congruent to angle B, then angle B is congruent to angle A.

Below is an example of how to use the symmetric property with *triangles*. Read the statement aloud. It states that if triangle X is congruent to triangle Y, then triangle Y is congruent to triangle X.

The symmetric property is similar to the substitution property in algebra.

**Transitive Property**

If we have two geometric figures that are equal to each other, and the second geometric figure is also equal to a third geometric figure, then the first geometric figure is equal to the third geometric figure.

Below is an example of how to use the transitive property with line segments. Read the statement aloud. It states that if line segment AB is congruent to line segment CD, and line segment CD is congruent to line segment EF, then line segment AB is congruent to line segment EF.

Below is an example of how to use the transitive property with *angles*. Read the statement aloud. It states that if angle A is congruent to angle B, and angle B is congruent to angle C, then angle A is congruent to angle C.

This geometric property is similar to the distributive property in algebra.

It is now time to take a break from learning about geometric properties and discuss the different types of angles that can be found in geometry. It is important to know the different types of angles in geometry. It is also important to know how to use deductive reasoning, because you will need to be able to write arguments to show that a particular theorem is true.

In geometry, there are four different types of angles. They are discussed below:

**Vertical angles**

As defined at the beginning of the lesson, vertical angles are two angles that are opposite each other where two lines intersect.

In the illustration above, angles A and B are vertical and angles C and D are vertical.

**Adjacent angles**

As defined in the beginning of the lesson, adjacent angles are two angles, in a plane, that share the same vertex or endpoint and the same side.

In the illustration above, note that there is a point (dot) in the middle of the four angles. That is the *endpoint*. Looking at the illustration, you can see that angles A and D are adjacent and angles C and D are adjacent. Can you name two more examples of adjacent angles from the illustration?

**Complementary angles**

As defined in the beginning of the lesson, complementary angles are two angles whose combined measurement is 90°.

**Supplementary angles**

As defined in the beginning of the lesson, supplementary angles are two angles whose combined measurement is 180°.

The next section of this lesson discusses a few of the theorems used in geometry. In this section of the lesson, you will learn how to use the information you presented earlier regarding how to make a convincing argument. It is not only important to learn and understand the theorems, you also need to know how to use them to prove a concept in a geometry math problem.

**Theorem 1-1: Vertical Angles Theorem**

This theorem states that vertical angles are congruent.

Suppose you are given the following geometry problem:

You are given that angle A and angle B are vertical angles. You must show that angle A and angle B are congruent.

- The first part of your argument should state the Angle Addition Postulate. (Refer to the Elephango lesson under
**Additional Resources**in the right-hand sidebar if you need assistance.) According to the Angle Addition Postulate, the measurement of angle A plus the measurement of angle D is 180°. In addition, the measurement of angle D plus the measurement of angle B is 180°. - The next part of your argument uses the substitution property. You can state that the measurement of angle A plus the measurement of angle D is equal to the measurement of angle B plus the measurement of angle D. If you subtract angle D from each side, then you are left with the measurement of angle A is equal to the measurement of angle B.
- The last part of your argument will simply state that, since the measurement of angle A is equal to the measurement of angle B, then the measurement of angle A is congruent to the measurement of angle B.

**Theorem 1-2: Congruent Supplements Theorem**

This theorem states that if two angles are supplements of congruent angles, then the two angles are congruent.

Suppose you are given the following geometry problem:

You are given that angle 1 and angle 2 are supplementary and angle 3 and angle 2 are supplementary. Prove (make a convincing argument) that angle 1 is congruent to angle 2.

- The first part of your argument uses the information that you are given in the problem.
- In the next part of your argument, all you need to do is state that if angle 2 were subtracted from each side of the equal sign, you are left with the measurement of angle 1 is equal to the measurement of angle 3.
- Your final part of your argument is to simply state that if angle 1 is equal to angle 3, then angle 1 is congruent to angle 3.

**Theorem 1-3: Congruent Complements Theorem**

This theorem states that if two angles are complements of congruent angles, then the two angles are congruent.

Suppose you are given the following geometry problem:

Given: Angles 1 and 2 are congruent. They are marked as such with the green arcs. Prove that angle 3 is congruent with angle 4.

- The first part of your argument is to state what you were given.
- The next part of your argument is to use the Angle Addition Postulate to show that angles 3 and 1 are supplements and angles 4 and 2 are supplements.
- The last part of your argument is to use the Congruent Supplements Theorem to state that angles 3 and 4 are congruent. Since angles 1 and 2 are congruent (you were given this information) and both angles 1 and 3 and angles 2 and 4 are supplements (proven with the Angle Addition Postulate), then it only stands to reason that angles 3 and 4 are congruent.

Although it may not be readily evident when you will use proofs in the real world, be assured that you will. Knowing how to complete a proof teaches you how to use deductive reasoning, as well as how to communicate clearly in all aspects of life.

The *Got It?* section offers activities for you to practice your new geometry skills. You can choose to complete one or both of the activities.

We help prepare learners for a future that cannot yet be defined. They must be ready for change, willing to learn and able to think critically. Elephango is designed to create lifelong learners who are ready for that rapidly changing future.

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