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!
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!
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(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:
This lesson will help you learn what deductive reasoning is and how to use it, and it 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:
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:
AB ≅ CD
The way to show that two angles are congruent to each other is below:
∠A ≅ ∠B
The way to show that two triangles are congruent to each other is below:
ΔX ≅ ΔY
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:
If AB ≅ CD, then CD ≅ AB
Reading 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:
If ∠A ≅ ∠B, then ∠B ≅ ∠A
Reading 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:
If ΔX ≅ ΔY, then ΔY ≅ ΔX
Reading 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:
If AB ≅ CD, and CD ≅ EF, then AB ≅ EF
Reading 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:
If ∠A ≅ ∠B, and ∠B ≅ ∠C, then ∠A ≅ ∠C
Reading 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 to 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 at 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.
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 were presented earlier regarding how to make a convincing argument. It is not only important to learn and understand the theorems, but 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.
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°.
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.
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.
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.
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.