Visual Guide to Triangle Construction

Constructing a triangle is a geometric algorithm application using only a Compass and a Straightedge. This article provides high-precision interactive simulations to visualize congruence postulates, specifically the technique of copying an angle.

Learn the basics here: Techniques for Geometric Construction: Lines and Angles.

1. Constructing Side - Side - Side (SSS)

This method is used when the lengths of all three sides ( $a, b, c$ ) are known. The vertex of the triangle is obtained from the intersection of two arcs centered at the endpoints of the base.

Input (px):

Construction Algorithm:

Inventory the known components (See the top area).
Draw Side c as the base (Line segment AB).
Measure Side b using the compass, draw an arc from A.
Measure Side a using the compass, draw an arc from B.
The intersection of the arcs is point C. Connect the lines.

2. Constructing Side - Angle - Side (SAS)

Precision is required when copying the angle. Observe the step where the "chord" (red arc) is measured to capture the angle's width.

Input:

Construction Algorithm:

Inventory components.
Draw Side 1 as the base.
Create an arbitrary arc on the Inventory angle, copy it to the base.
Precision: Measure the chord width (distance between legs), intersect it on the base arc.
Draw a ray, then mark off Side 2.

3. Constructing Angle - Side - Angle (ASA)

This process involves copying an angle twice. The intersection of the two rays will automatically form the triangle's vertex.

Input:

Construction Algorithm:

Inventory components.
Draw the base line.
Copy Angle A to the left endpoint (measure arc & chord).
Copy Angle B to the right endpoint (measure arc & chord).
The intersection of rays is C.

4. Constructing Side - Angle - Angle (SAA/AAS)

In SAA (or AAS), we are given a side and two angles, but the side is not included between the angles. To construct this, we must first find the third angle using the property that the sum of angles in a triangle is 180°.

Input:

Construction Algorithm:

Inventory: Side, Adjacent Angle ( $α$ ), Opposite Angle ( $β$ ).
Step 1: Calculate the 3rd Angle ( $γ$ ) in the Inventory using a 180° line.
Step 2: Draw the Side as the base.
Copy the Adjacent Angle ( $α$ ) to one end.
Copy the Calculated 3rd Angle ( $γ$ ) to the other end. (Using ASA method).