When you’re building a house, the most important part of that house is its foundation. The foundation helps you build the layers and ultimately create a home!
Geometry is quite similar to building a house. In this unit, we will discover the most important pieces of the foundations of Geometry. Afterwards, we will discover how these foundations can transform into patterns and properties to discover!
To begin, there are vocabulary words in Geometry that we call undefined terms. These terms are words that we can’t necessarily define, but we know how to picture them and name them:
To begin, there are vocabulary words in Geometry that we call undefined terms. These terms are words that we can’t necessarily define, but we know how to picture them and name them:
Point Line Plane |
This is called point \(C\)
This line can be called line \(m\), \(\overleftrightarrow{AB}\) or \(\overleftrightarrow{BA}\).
This plane can be called plane \(HAK\) (the letters can be in any order), or plane \(\mathcal{M}\).
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To name a point, we usually label each point with a single capital letter.
There are two ways to name a line: by using a single lower case letter labeling near one of the arrows at the end, or by naming two points on the line (order doesn’t matter) and placing a line symbol on top. There are two ways to name a plane: by using a single cursive upper case letter in the corner or by using three non-collinear points that are on the plane. |
With these undefined terms we can define other terms that are crucial in our understanding of Geometry.
Segment
Ray Opposite Rays Angle Intersection Coplanar Collinear |
The part of a line that connects two points (endpoints).
The part of a line that starts from one point (initial point) and extends in one direction from that point. Also called a half line. Two rays sharing an initial point that go in opposite directions. They must form a straight line. Two rays (sides) that share the same initial point (vertex). Describes the point(s) two figures have in common. Points, lines, or segments that lie in the same plane. Points that lie on the same plane and the same line. |
This segment can be called \(\overline{JM}\) or \(\overline{MJ}\).
This ray is called \(\overrightarrow{KJ}\).
\(\overrightarrow{EA}\) and \(\overrightarrow{EB}\) are opposite rays since they form a straight line.
This angle can be called \(\angle {A}\), \(\angle 1\), \(\angle{CAB}\), or \(\angle{BAC}\).
The intersection between line \(m\) and line \(n\) is point \(A\).
The intersection between plane \(\mathcal {M}\) and plane \(\mathcal{N}\) is \(\overleftrightarrow {GE}\) or line \(r\). Points \(M, N, R, T\) and \(S\) are coplanar because they all lie on plane \(\mathcal {E}\).
Line \(a\) and line \(b\) are coplanar because they lie on plane \(\mathcal {F}\). Points \(C, A,\) and \(E\) are collinear because they lie on the same line.
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To name a segment, use the two points at the ends of each segment (called endpoints). Note: the order of the endpoints doesn’t matter.
To name a ray, the order of the letters matter. You must start with the initial point first! Opposite rays are made up of two rays. So you will name it the same way you would name rays (see above). To name an angle, use the following symbol: \(\angle\) or \(\measuredangle\). When the angle is alone, you can name it by its vertex. Otherwise, use three letters with the vertex is in the middle. Or, you can name an angle with a number that is the vertex where the two rays meet. Depending on the situation, intersections can be points, lines, segments, or planes! Since we can talk about coplanar points, coplanar segments, and coplanar lines - we will name them as we normally name points segments, and lines. Name these as points! |
Geometry developed in China, Egypt, Babylonia, India and Greece over many centuries. The Greeks developed an axiomatic system , principles of Geometry that everything is derived from. A postulate (also known as an axiom) is something that is accepted without proof. Mathematicians put postulates or axioms together to prove new ideas called theorems. The first collection of this axiomatic system was written by the Greek mathematician Euclid in his book called the Elements around 300 BC. The Elements is the basis of all Geometry textbooks, including the Digi!
An excerpt of the Elements is below.
An excerpt of the Elements is below.
All of these figures can be constructed and manipulated with a compass and straightedge on paper or using dynamic geometry software such as Desmos and GeoGebra. Move the figures around and see what happens.
A static image of intersecting planes can be difficult to picture as a 3D object. Below is an interactive app of intersecting planes. Figure out which points move the plane and which points do not.
With all of this new vocabulary, let’s see if you can identify and correctly label each of them through the example below.
Example 1: Use the diagram to determine if the statement is true or false. If the statement is false, correct it so that it is true!
Example 1: Use the diagram to determine if the statement is true or false. If the statement is false, correct it so that it is true!
Solutions:
a) False. Although points \(P, R,\) and \(U\) are on the same plane, they do not lie on the same line. Thus they cannot be collinear. To make the statement true, we can change point \(P\) to point \(Q\): Points \(Q, R,\) and \(U\) are collinear.
b) False. Even though point \(R\) is the vertex of \(\angle 1\), this is what we call the ambiguous case. If you say \(\angle R\), you could be talking about many different angles whose vertex is (R\). Thus, we MUST use three letters to describe \(\angle 1\): \(\angle TRU\) or \(\angle URT\).
c) False. Planes \(PUR\) and \(RMN\) have more than one point in common - they have all the points on \(\overleftrightarrow{TR}\) (or line \(b\)) in common! Thus the intersection of plane \(PUR\) and plane \(RMN\) is \(\overleftrightarrow{TR}\).
d) False. Opposite rays must point in opposite directions, but \(\overrightarrow{QR}\) and \(\overrightarrow{RU}\) point in the same direction. To make the statement true, we can change it to say: \(\overrightarrow{RQ}\) and \(\overrightarrow{RU}\) are opposite rays.
e) True. You can use any three points that are on plane e to describe it. Since points \(M, T,\) and \(N\) are on plane \(\mathcal {E}\), this is one valid way to name the plane!
f) True. Since points \(Q, R,\) and \(U\) are collinear, you can use any pair of those three points to describe the line.
Example 2: Use the diagram to answer each question.
a) False. Although points \(P, R,\) and \(U\) are on the same plane, they do not lie on the same line. Thus they cannot be collinear. To make the statement true, we can change point \(P\) to point \(Q\): Points \(Q, R,\) and \(U\) are collinear.
b) False. Even though point \(R\) is the vertex of \(\angle 1\), this is what we call the ambiguous case. If you say \(\angle R\), you could be talking about many different angles whose vertex is (R\). Thus, we MUST use three letters to describe \(\angle 1\): \(\angle TRU\) or \(\angle URT\).
c) False. Planes \(PUR\) and \(RMN\) have more than one point in common - they have all the points on \(\overleftrightarrow{TR}\) (or line \(b\)) in common! Thus the intersection of plane \(PUR\) and plane \(RMN\) is \(\overleftrightarrow{TR}\).
d) False. Opposite rays must point in opposite directions, but \(\overrightarrow{QR}\) and \(\overrightarrow{RU}\) point in the same direction. To make the statement true, we can change it to say: \(\overrightarrow{RQ}\) and \(\overrightarrow{RU}\) are opposite rays.
e) True. You can use any three points that are on plane e to describe it. Since points \(M, T,\) and \(N\) are on plane \(\mathcal {E}\), this is one valid way to name the plane!
f) True. Since points \(Q, R,\) and \(U\) are collinear, you can use any pair of those three points to describe the line.
Example 2: Use the diagram to answer each question.
a) Name \(\angle 2\) in as many ways as possible.
b) Name two pairs of opposite rays.
c) Describe the intersection between line \(m\) and line \(n\).
Solution:
a) Let's watch the video for the solution.
b) Name two pairs of opposite rays.
c) Describe the intersection between line \(m\) and line \(n\).
Solution:
a) Let's watch the video for the solution.
b) \(\overrightarrow{AC}\) and \(\overrightarrow{AE}\) are opposite rays since they form a straight line and are pointing in opposite directions. Another pair of opposite rays is \(\overrightarrow{AB}\) and \(\overrightarrow{AD}\).
c) The intersection between line \(m\) and line \(n\) is point \(A\), since that is the only point that the two lines have in common!
Extension Solving Systems of Equations
The intersection of two lines is a point. This point is a solution for both equations of lines. Click here to review solving systems of equations.
c) The intersection between line \(m\) and line \(n\) is point \(A\), since that is the only point that the two lines have in common!
Extension Solving Systems of Equations
The intersection of two lines is a point. This point is a solution for both equations of lines. Click here to review solving systems of equations.
Quick Check
1) Name \(\angle 1\) in as many ways as possible.
1) Name \(\angle 1\) in as many ways as possible.
2) Match each image with the appropriate term.
a) Segment
b) Line
c) Ray
b) Line
c) Ray
3) Use symbols to name each of the figures below
4) Use the diagram to answer the questions that follow.
a) Name another point that is coplanar to points \(G\) and \(E\).
b) Describe the intersection between plane \(\mathcal {M}\). and plane \(\mathcal {N}\).
Quick Check Solutions
b) Describe the intersection between plane \(\mathcal {M}\). and plane \(\mathcal {N}\).
Quick Check Solutions