Constructing Triangles

SSS (Side-Side-Side)

• Step 1: Draw the longest side as a horizontal base using a ruler.
• Step 2: Set the compass to the length of the second side, place the point on one end of the base, and draw an arc.
• Step 3: Set the compass to the length of the third side, place the point on the other end of the base, and draw a second arc that intersects the first.

SAS (Side-Angle-Side)

• Step 1: Draw one of the given sides as the base.
• Step 2: Use a protractor to measure the required angle from one endpoint of the base and draw a faint construction line.
• Step 3: Use a ruler to measure the second side length along this construction line and connect the endpoint to the other side of the base.

ASA (Angle-Side-Angle)

• Step 1: Draw the given side as the base.
• Step 2: Measure and draw the first angle from one end of the base and the second angle from the other end.
• Step 3: Extend both lines until they intersect to form the third vertex.

• SSS vs. SAS: SSS relies entirely on side lengths and requires a compass for accuracy, whereas SAS requires a protractor to establish the fixed opening between two sides.

• ASA vs. AAS: While ASA provides the side between two angles, AAS (Angle-Angle-Side) requires calculating the third angle using the $180^\circ$ rule before construction can begin on a single base line.

• Never Erase Arcs: Examiners look for construction arcs as evidence of the correct methodology; erasing them may result in a loss of marks even if the triangle looks correct.

• Check the Scale: When using a protractor, always ensure you are reading from the correct zero-line (inner vs. outer scale) to avoid drawing obtuse angles instead of acute ones.

• Accuracy Check: After construction, measure the remaining sides or angles to ensure they match the theoretical values; for example, in an ASA triangle, the third angle should satisfy the $180^\circ$ rule.

• Sharpness Matters: Use a very sharp pencil for both the ruler lines and the compass arcs to minimize the margin of error, which is often strictly penalized in geometry exams.