In the first lessons, "construct" meant that you are allowed to use only a compass and the straight edge (not the markings) of a ruler. There are other sets of tools that are commonly (and uncommonly) used for geometric constructions.
You are allowed to use markings on the ruler and protractor to draw any lengths and any angles. You can use the straight edge to draw lines.
You are allowed to arrange toothpicks in a straight line (use a ruler to make sure you don't cheat and get subtly curved lines). You may not break toothpicks. Thus, with toothpicks, you can construct line segments with integer lengths only.
Get plastic drinking straws and paper clips that just fit into the straws. To connect two (or more) straws, hook two (or more) paper clips together at the same point, then stick each paper clip into the end of a straw. (Exercise: Make an equilateral triangle with straws. How many straws and clips will you need?) The angles you make with paper clips at the vertices are adjustable.
Option: Adjustable lengths. Flatten and fold another straw and insert it into the end of an unflattened straw. You can change the length of the combined straws by how far in the flattened straw is.
Option: Straws, paper clips, and ruler. With these rules, you can cut straws to the lengths you want.
Option: Straws, paper clips, and protractor. With these rules, you can fix angles at the measure you want. To construct a fixed 45o angle, for example, draw a 45o angle on paper using a protractor. Cut out the angle and tape a straw-and-paper clip joint onto the angle.
Definitions: A triangle is equilateral if all its sides
have the same length.
A triangle is isosceles if two of its sides have the same length.
A triangle is scalene if none its sides have the same length.
Definitions: An angle is obtuse if its measure is more
than 90o (and less than 180o).
An angle is right if its measure is 90o.
An angle is acute if its measure is less than 90o (and
more than 0o).
Definitions: A triangle is obtuse if one of its angles
is obtuse. (What happens if more than one angle is obtuse?)
A triangle is right if one of its angles is right. (What happens
if more than one angle is right?)
A triangle is acute if all of its angles are acute.
Definitions: Two sides of a polygon are adjacent if they
share a vertex.
The included angle between two adjacent sides of a polygon is the
angle formed by the two sides.
The included side between two consecutive angles of a polygon is
the side that is shared by the two angles.
An angle and a side are opposite in a triangle if the side is not
part of the angle.
Definitions: Two geometric objects are congruent if they have the same shape and size. This means that one of the objects can be moved (including reflecting) so that it coincides with the other. If two objects are congruent, then all corresponding measurements of the objects will be equal. This includes all kinds of length measurement (height, width, perimeter, radius, circumference, etc.), angles, areas, and volumes.
Two geometric objects are similar if they have the same shape but not necessarily the same size. This means that one of the objects can be uniformly stretched or shrunk so that it is congruent to the other. If two objects are similar, then all corresponding length measurements of the triangles will be proportional, and all corresponding angle measurements are equal. (We will discuss later the relationships between areas and volumes.)
From your list, formulate a rule that allows you to determine whether a triangle exists with three given lengths for sides. Your rule should work for non-integer lengths, too.
Is there a triangle with sides of lengths 37, 23, 65?
When all groups are finished, group the whole class's triangles by "Type of measurements":
For each type, make a conjecture like this: "If two sides and an included angle of a triangle are given, then all [or not all] triangles having those measurements are congruent [or similar]." Be as specific as you can.