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Vertex and Diagonals - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The plural form of vertex, vertices are typically intersections or corners in figures. In an angle, the vertex is where two lines, rays, or segments meet. A diagonal is a line segment whose endpoints are vertices. In a polygon, a vertex is where two edges or sides meet. In a polyhedron, a vertex is where three edges meet.

A word that's used throughout Geometry is Vertex. If you have just one it's Vertex, if you have more than one that is if you want to make it plural it's vertices. So one vertex, two vertices and a vertex shows up in lots of different areas.
If we're talking about angles for example; the vertex is the endpoint of the two rays. So here point b would be our vertex, is where ray ba and ray bc where they began forming the sides of this angle.
What about triangles? If you have an isosceles triangle, you have two legs which I guess I could label so have a leg here and leg here and this angle formed by the two legs is called your vertex angle so I'm going to call this my vertex angle. So vertex could also apply to isosceles triangles.
What about other types of polygons? Well the point where two sides meet which I've written in blue here, so this point right here is where two sides have intersected. So we have five vertice- vertices in this polygon. A diagonal is related to a vertex because a diagonal is a line segment that connects any two non-consecutive vertices. Well consecutive vertices, if I look at this vertex its consecutive vertices are the one to its right and the one to its left. So a diagonal from this vertex would connect these two vertices which are not consecutive. I can also draw one in to one other non-consecutive vertex. So that's how a diagonal is related to a vertex.
If we're talking about three dimensions, we can look at a cone or a pyramid and this point right here in the cone and the point in the pyramid is also known as the vertex. So a vertex can apply to 3 dimensional solids.
What about a cube? Well in a cube, or any type of rectangular prism, is a point where three or more edges intersect. So the edges are these line segments where two planes intersect so a vertex could be right here right here you're going to have in this cube, if I drew in the rest of it the hidden sides and hidden edges you're going to have 1, 2, 3, 4, 5, 6, 7, 8 total vertices. Now again if you had a hexagonal prism, then you'll have a different number of vertices but a vertex, when you're talking about a prism is the point where three more edges intersect.
So you can use vertex in lots of different ways; in prisms, pyramids, cones, polygons, isosceles triangles and angles. So it's the same word we're just applying it to different situations.

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