Constructions

Objective

• Go over some tools you will need for Geometry
• Go over what a construction is
• Go over why we need to know how to do them
• Go over some parts of angles
• Go over some different types of angles
• Go over how to use a protractor to make a certain angle

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So to start with, what do we need?

What we need to start a proper construction is a few things.

We need a:

Compass

Protractor

Straight Edge:

(The underside of a protractor may be uses as well)

Alright, we have what we need, but what is a construction?

Well, according to Google, a construction is:

“The style of method used in the building of something”

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Which in this case is pretty spot on.

So what we’re going to be doing for right now is basically creating different angles.

We’re doing this because this is Geometry, which loosely translates to:

“measuring the world”

So why do we need to measure angles in order to measure the world?

Because every shape that you can think of can be thought of as a bunch of angles put together.

(Including round shapes, which deal with what we call arcs).

So, to start off, let’s begin by showing how to construct an angle of certain measure:

Creating a certain angle:

So, the first thing we need to create a specific angle is our straight edge, and protractor.

So, first we take our straight edge:

And we draw a straight line segment:

Now, we need our protractor

We place the center of the protractor at one of the endpoints of our line segment

We mark the certain angle that we want our angle to be:

(in this case, 60 degrees)

And now we connect the endpoint of our segment, with the angle point we made:

And we have our angle:

Important definitions:

Line: a line is a figure that is straight (doesn’t bend), has no thickness, and extends in both directions without end.

It is represented as:

Ray: a ray is a figure that starts at an endpoint, and continues on forever in one direction.

Think of it like half of a line:

Line segment: a line segment is a figure that has two endpoints.

What we normally call a line, is usually a line segment.

Angle: an angle is a figure that is formed by two rays sharing the same end point.

End Point: an end point is a point where the line segment, angle, or ray, stops.

Vertex: the vertex of an angle is the end point of the angle.

Side of the angle: the sides of the angle are the two rays that make up the angle.

HOW TO COPY AN ANGLE

So, now that we’ve learned the important vocab, and how to use our tools.

It’s time to put them to use.

So, let’s copy an angle.

First, we’re going to need our compass and straight edge.

Next, we need an angle and a point outside of the angle

something like this:

Now, to copy our angle, we are going to take our compass:

And we’re going to measure a portion of one of the sides of the angle:

Now, without changing our measurement,

We’re going to take our compass, and move it to our point.

Then we’re going to make another arc

the same way we did it the first time.

Now, we’re going to measure the points we made on our first marking

And with our compass, we’re going to measure the distance in between those points

Now we’re going to move our compass to the intersection of the arc and the other ray.

Then we make a ray from the point.

And we’ll make our mark.

Now make a ray that passes through the intersection:

And now we’ve copied the angle perfectly!

GREAT, SO WHY DO WE NEED TO KNOW THIS?

Well, mainly because the other way we would do it is by using a ruler to measure.

But not all rulers are exact. �They’re really really close!

But this creates a more exact copy of an angle without the need for a ruler to measure it out.

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So now that we’ve gone over how to copy an angle, what about finding an angle bisector?

Well, first let’s go over what a bisector is:

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A bisector can be defined as a line that separates something into two equal parts.

Basically, it cuts it in half.

So, now that we’re a little more familiar with how to use a compass and all.

Let’s get into how to create a segment bisector.

HOW TO CREATE A SEGMENT BISECTOR

First, like before, we’re going to need our compass and straight edge.

Next, we need a segment to work on.

Something like this:

Now, to essentially cut our segment in half

(Without measure it)

We’re going to first pick a point that is further than the middle of the segment.

Then, we’re going to make our mark:

Now, we’re going to do the same thing, but this time on the other side.

Now place a point where the two arcs intersect.

And now we drop a segment from our point to the line:

And right where our two segments intersect is the midpoint, or middle point/halfway point, of our segment.

HOW TO CREATE AN ANGLE BISECTOR

First, like before, we’re going to need our compass and straight edge.

Next, we need an angle to work on.

Something like this:

Now, to essentially cut our angle in half

(Without measure it)

We’re going to first place our compass on the vertex of the angle.

Then we’re going to make an arc.

Now, without changing the measurement, we’re going to find the intersection of our arc and our ray.

And draw another arc

Then we’re going to do it again.

So, without changing the measurement, we’re going to find the intersection of our arc and our ray.

And draw another arc

Now, the point of intersection is what we are looking for.

So, now we’re going to draw a ray from the vertex of our angle

To the point of intersection

Like so:

And there is our angle bisector.