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Congruence, Proof, and Constructions

Definitions of Geometric Objects

The most basic objects used in geometry are points, lines, and planes. However, these objects are called undefined terms since they do not have a precise definition — they are described using their characteristics. But despite this, their descriptions can be used to define many more geometric objects properly.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Categorizing Geometric Objects

Tearrik's baby accidentally knocked some boxes off the shelf. Now, the objects are cluttered all over the floor.
Four boxes labeled with Segment, Ray, Angle, Circle. Student needs to move each object shown to the corresponding box
Help Tearrik clean up the mess by moving each object to the corresponding box.

Discussion

How Are Angles Formed?

When having two distinct rays on the same plane, there are various ways of positioning them. If it were to be the case that both rays have the same starting point, the geometric figure has its own name, an angle.

Concept

Angle

An angle is the set of points in the plane formed by two different rays that share the same starting point. This common point is called the vertex of the angle and the rays are the sides of the angle.

Two Rays with the same starting point

There are different ways to denote an angle and all involve the symbol in front of the name. For the second notation, the vertex always must be in the middle.

Using the Vertex Using the Vertex and One Point on Each Ray Using a Number
or

The measure of an angle, denoted by is the number of degrees between the rays. It is found by applying the Protractor Postulate. When two angles have the same measure, they are said to be congruent.

Extra

Interior and Exterior of an Angle

An angle divides the plane into two parts.

  • The region between the sides, or interior of the angle.
  • The region outside the sides, or exterior of the angle.
These regions can be examined in the following graph.
Interior and Exterior of an angle
Notice that the interior of an angle is the region for which the angle measure is less than

Discussion

An Angle Defined Using Transformations

Consider angle What type of movement could be applied to such that it lies exactly on
Investigating if a translation or rotation sends a ray onto other
A rotation around can send onto This fact allows for an angle to be defined in terms of transformations.

An angle is a figure formed by a ray and its image after being rotated around its endpoint.

Using this definition, the measure of the angle is the angle of rotation that maps one ray onto the other.

Example

Students Define Angles

At a geometry forum, one question made by the audience was to say what an angle is. The only ones who answered this question were Heichi and his friends.

a Heichi said an angle is the set of points formed by two different rays.
b Emily said an angle is the set of points formed by two different rays with a common point.
c Kevin said an angle is the set of points formed by two different rays with the same endpoint.

Analyze each definition, indicate whether it is mathematically correct, and point out the flaws, if any.

Answer

a Heichi's answer has a major flaw. It includes incorrect cases, for instance, when the rays do not intersect.
b Emily's answer has the flaw of not being precise. It includes incorrect cases, for instance, two intersecting rays with different endpoints.
c Kevin's answer is correct and the most classical definition.

Hint

a Consider all possible positions for two different rays in a plane.
b Notice that it is not said whether the common point is the endpoint of both rays.
c For two rays, having the same endpoint is the same as sharing the same starting point.

Solution

a Heichi's answer has a major flaw. He did not mention any relationship between the rays other than being different. Then, his definition includes the following four cases.
  1. Two non-intersecting rays.
  2. Two intersecting rays with different endpoints.
  3. Two rays such that one of them passes through the endpoint of the other.
  4. Two rays with the same endpoint.
Different positions for two different rays on a plane

Among these cases, only the fourth one corresponds to an angle. Consequently, Heichi's definition is not precise since it includes incorrect cases.

b As with Heichi, Emily's answer has the flaw of not being precise. Besides the fourth and correct case, her answer includes the second and third cases, which are not correct.
c Kevin's answer is correct. Although he said that both rays have the same endpoint, this is equivalent to say that both rays share the same starting point.

Discussion

Definition of a Circle

Consider two points and on a plane such that their distance is Apart from are there more points on the plane whose distance to is also

Two points 2 units apart

The answer is yes. There are infinitely many, and the set of all these points is called a circle.

Concept

Circle

A circle is the set of all the points in a plane that are equidistant from a given point. There are a few particularly notable features of a circle.

  • Center - The given point from which all points of the circle are equidistant. Circles are often named by their center point.
  • Radius - A segment that connects the center and any point on the circle. Its length is usually represented algebraically by
  • Diameter - A segment whose endpoints are on the circle and passes through the center. Its length is usually represented algebraically by
  • Circumference - The perimeter of a circle, usually represented algebraically by
In the applet, the center is labeled Therefore, the circle can be referred to as or circle
parts of a circle
In any given circle, the lengths of any radius and any diameter are constant. They are called the radius and the diameter of the circle, respectively.

Discussion

Circle Definition Using Transformations

A circle can also be defined as the path that a point leaves when it is rotated about a different point called the center.
A circle made by rotating a point around another point
The radius of the circle is the distance between the center and the initial point.

Example

A Circle and Its Center

Let be a circle centered at point with a radius of centimeters. Pick any point on this circle and label it as

a Does belong to
b Consider such that it has a radius of centimeters. What is the relationship between and this circle?

Answer

a No
b Point belongs to the circle centered at point with a radius of centimeters.

Hint

a A circle is the set of all points that are equidistant from a given point.
b Consider the distance between and

Solution

a By definition, a circle is the set of all points that are equidistant from a given point. In this case, is the set of points whose distance to is centimeters.
A circle centered at P a radius 3

Since point is not centimeters apart from itself, it does not belong to

b By definition, the circle centered at with a radius of centimeters is the set of all points whose distance to is centimeters. Since belongs to its distance to is centimeters. In consequence, point belongs to the circle centered at
A circle centered at P and a circle centered at Q, both with the same radius

Explore

Positioning Lines in a Plane

When two lines are drawn in a plane, there are two possible options — they intersect each other or they do not intersect at all. Move the points and to investigate the name given to the lines when they do not intersect. Is there only one way to position and such that and do not intersect?
Two free points to draw a line parallel to a given line
For the next definition, it is crucial that the lines are in the same plane.

Discussion

Parallel Lines

Two lines are parallel lines when they do not intersect and are coplanar. In a diagram, to denote that two lines are parallel, triangular hatch marks are drawn on the lines.

One pair of parallel lines and one transversal
To write algebraically that two lines are parallel, the symbol is used. In the diagram, whereas is neither parallel to nor

Discussion

Parallel Lines Using Transformations

Consider a pair of parallel lines. Is it possible to translate one line such that it lies exactly on the other? Try moving line onto line
Moving one line onto another parallel line
The diagram confirms that it is possible to send one line onto the other using a translation. This allows defining parallel lines in terms of transformations.

Two lines are parallel when one of the lines is the image of the other under a translation.

Notice that this definition does not require the lines to be coplanar.

Example

Students Define Parallel Lines

Last week, Kevin and his friends learned about parallel lines. Today, the teacher asked them to define, without using their notes, what does it mean for two distinct lines to be parallel.

a Kevin said two distinct lines are parallel when they have the same slope and are coplanar.
b Emily said two distinct lines are parallel when they do not meet.
c Heichi said two distinct lines are parallel when they do not intersect each other and are coplanar.

Analyze each definition and indicate whether it is mathematically correct. Point out the flaws, if any.

Answer

a Kevin's answer has two flaws. First, it does not cover the case when one of the lines is vertical, and therefore has no slope. Secondly, it refers to the slope of a line, which is not essential for the formal definition.
b Emily's answer is not precise and includes the case of non-coplanar lines, which is incorrect. This can be fixed by adding that the lines are coplanar.
c Heichi's answer is correct and the most classical definition.

Hint

a What is the slope of a vertical line? Is the slope strictly needed when talking about parallel lines?
b Think about two non-coplanar lines that do not intersect. Are they necessarily parallel?
c Remember the two conditions for two lines to be parallel.

Solution

a Once again, Kevin gave a definition in terms of the slope of a line. This is a flaw since it only applies to lines having a slope. This definition does not tell whether or not a pair of vertical lines are parallel or whether or not a vertical line is parallel to a non-vertical line.

Vertical lines have no slope.

Consequently, Kevin's definition fails to cover all possible cases. However, this flaw can be corrected by adding the following premise.

All vertical lines are parallel and no vertical line is parallel to a non-vertical line.

The second flaw in Kevin's definition is that it refers to auxiliary information — the slope of a line. Whereas it is true that two non-vertical lines are parallel when they have the same slope, it is possible to define parallel lines without referencing the slope.

b Emily's answer seems correct, however, she did not say whether the lines are coplanar or not. Without this condition, two lines might not intersect each other nor be parallel. Such is the case for skew lines.
Two skew lines

Consequently, Emily's definition is not precise and therefore includes incorrect cases. Her statement can be fixed by adding that the lines are coplanar.

c Notice that Heichi's answer is the same as Emily's answer, but he included that the lines are coplanar. Therefore, Heichi's definition is correct and would be considered the most classical definition.

Explore

What Makes Two Lines Perpendicular?

Whenever two coplanar lines intersect at a single point, they form four angles. By moving points and investigate the measure of the four angles that make the lines and perpendicular. Is there only one line perpendicular to
Two free points to draw a line perpendicular to a given line
Despite the previous diagram shows four angle measures, knowing only one angle measure is enough to define perpendicular lines.

Discussion

Perpendicular Lines

Two lines are perpendicular lines when they intersect at a right angle and are coplanar. In the diagram, is perpendicular to and is expressed as

Two lines intersecting at a right angle

Discussion

Perpendicular Lines Using Transformations

Notice that even knowing only one angle measure, it is possible to know if two lines are perpendicular. Consider a pair of perpendicular lines and Is there a way to map line onto line by applying a rotation about the movable point Try it out!
Rotation of line L around point C
From the exploration, it can be seen that a rotation of around the intersection point sends onto This allows defining perpendicular lines in terms of transformations.

Two lines are perpendicular when one of the lines is the image of the other under a rotation of around the intersection point.

Example

Students Define Perpendicular Lines

Three students were asked to write down what it means for two distinct lines to be perpendicular. Analyze each of their answers and indicate whether they are mathematically correct. Point out the flaws, if any.

a Kevin wrote, two distinct lines are perpendicular when the product of their slopes is
If the product of the slopes is -1, the lines are perpendicular
b Heichi answered, two distinct lines are perpendicular when they form four angles of equal measure.
Two intersecting lines forming four congruent angles
c Emily wrote, two distinct lines are perpendicular when they intersect at a angle.
Two intersecting lines forming a 90 degree angle

Answer

a Kevin's answer has two flaws. First, it does not cover the case when one of the lines is vertical, and therefore has no slope. Secondly, it refers to the slope of a line, which is not essential for the formal definition.
b Heichi's answer is correct. However, it requires finding four angle measures when knowing only one is enough.
c Emily's answer is correct and the most classical definition.

Hint

a What is the slope of a vertical line? When talking about perpendicular lines, is it essential to mention the slope?
b The four angles formed by two intersecting lines make a complete turn of Are the four angle measures strictly needed?
c A angle is the same as a right angle.

Solution

a Kevin's answer has two flaws. First, it only applies to lines that have a slope. Since vertical lines have no slope, Kevin's answer does not tell whether a vertical line can be perpendicular to another line. Then, his definition fails to cover all possible cases. Adding the following premise would correct this flaw.

Vertical lines are perpendicular to horizontal lines.

Secondly, Kevin's answer refers to auxiliary information — the slope of a line. While it is true that two non-vertical lines are perpendicular when the product of their slopes is it is possible to define perpendicular lines without referencing the slope.

b Consider two lines that intersect and form four angles of equal measure.
Two intersecting lines forming four congruent angles

Since a complete turn is each angle must have a measure of That is, the lines form four right angles, and consequently, Heichi's answer is correct. However, his answer has a minor flaw. It requires finding four angle measures when only one is enough.

c Since a angle is the same as a right angle, two lines are indeed perpendicular when they intersect at a angle. Therefore, Emily's answer is correct and the most classical definition. Notice that, in contrast to Heichi's answer, she only mentioned one angle measure.

Discussion

Defining a Line Segment

One last geometric object to be considered is the section or part of a line between any two of its points.

Concept

Line Segment

A segment, or line segment, is part of a line bounded by two different points, called endpoints. It is made of all the points on the line between the endpoints. Unlike a line, a segment does not extend infinitely, and then it is drawn without arrowheads.

Segment from A to B
As with a line, between two different points there is exactly one segment. A segment with endpoints and is denoted as or while its length or measure is written as or When two segments have the same length they are said to be congruent.

Example

Students Define Line Segments

In the final of a math Olympics, two students from different schools had to define what a line segment is.

a LaShay, from Jefferson High, said a line segment is a portion of a line bounded by two different points and contains all the points in between.
b Mark, from South High School, said a line segment is the intersection of two different rays that are on the same line but have opposite directions.
Analyze each answer and indicate whether it is mathematically correct or not. Which school should win the Olympics?

Answer

Jefferson High should win the Olympics.

a LaShay's answer is correct and the most classical definition.
b Mark's answer is not precise. It includes cases where the intersection is not a segment. For instance, when the rays are opposite rays or when the rays do not intersect at all.

Hint

a Remember that a segment has a length and in consequence, it must be bounded.
b Consider all possible positions for two rays that lie on the same line but have opposite directions.

Solution

a LaShay's answer is correct and is the classical definition of line segment.
b Mark's answer has a major flaw. It does not specify a particular position for the rays. Therefore, it includes the three following cases.
  • When the rays are opposite rays. Here, their intersection is only the endpoint.
    Different positions for two rays on the same line with opposite directions
  • When the rays do not intersect at all.
    Different positions for two rays on the same line with opposite directions
  • When the rays intersect at more than one point. Here, the intersection is a line segment.
    Different positions for two rays on the same line with opposite directions

Since the first and second cases do not represent a line segment, Mark's definition is not correct. It seems like Mark did not consider the first two cases. In consequence, Jefferson High should win the Olympics.

Closure

Real-World Modeling with Geometry

Consider a pair of parallel lines and and a line which is perpendicular to both lines. No matter where the line is placed, the distance between the two points of intersection is always the same as the distance between the lines.
Parallel lines and one perpendicular
The previous fact can be used in different real-world scenarios. For example, think about a train. The distance between the wheels in each axle is always the same. Therefore, the rails must always be a specific distance from each other; otherwise, at some point, the train could crash.
Error loading image.

Consequently, to avoid crashing, the rails must be parallel to each other. Thus, the railroad tracks can be viewed as a pair of parallel lines. Here, the train's axles and the sleepers are perpendicular to each rail.

Railroad
Be aware that the real world is not as perfect as the scenarios considered in mathematics. For instance, in the real world, lines do not extend infinitely, and planes have thickness. However, these perfect examples are studied because they allow for the ability to draw logical conclusions and estimate real-world cases very well.
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