How many ways can you bring end-points G and H together to form a triangle. Given that their lengths are all constrained, what does that imply about the angles?[br][br]If the lengths are not constrained, are their any limits on the size of the angles or sides when end-points C and D are brought together?[br][br]What if the three angles are fixed? For instance if they are all 60 degrees? Or 30, 60 and 90 degrees?[br][br]Can there be a combination of some side lengths and some angle sizes that determine a fixed triangle?

When you created triangle DEF, was it congruent to ABC? How could you tell?[br]Can you state a theorem (a provable rule) that summarizes what you discovered?[br][br]In some geometry books, this is called the "Side-side-side" (SSS) rule:[br]If two triangles have the same three side lengths, then the triangles are congruent.[br]Many conclusions in geometry can be proven using this theorem.

Can you recreate this pair of triangles: any triangle ABC and another triangle that has one angle and the two sides forming that angle congruent to the corresponding parts in ABC?[br]Can you drag those triangles to show that they are congruent and remain congruent no matter how triangle ABC and its vertices are dragged?[br][br]The theorem you have explored is called "Side-Angle-Side" or "SAS".[br][br]Are two triangles necessarily congruent if they have one angle and two sides congruent, but the angle is not between the two sides?

Can you copy segment AB and the angles at A and B to a new segment?[br]Make a polygon connecting the intersection of the sides with the two copied vertices.[br]How is the new triangle dependent on the original? Are the two triangles constrained to be congruent?[br][br]There is a theorem called "Angle-Side-Angle" or "ASA" that says that if two triangles have two angles and the included side congruent, then the two triangles are congruent.[br][br]Do you see that this is always true as you drag the vertices of the original triangle?[br][br]Does this mean that if two triangles have two angles and any side equal, then the two triangles are congruent?[br][br]Note that the three angles of a triangle always add up to 180 degrees. So if two of the angles are fixed, then so is the third (180 minus the sum of the other two angles). Does this mean that two angles and any side will determine a congruent triangle?

Can you

Could you construct the two triangles?[br]When is it possible to construct two different triangles with SSA fixed?[br][br]What combinations of congruent sides and/or angles determine congruent triangles?[br]E.g., SSS and SAS, but not SSA.

Triangle DEF is "inscribed" in triangle ABC. This means that DEF fits exactly inside of ABC, with each vertex of DEF on a side of ABC.[br][br]You know how to construct an equilateral triangle like ABC from Challenge 7. What happens when you try to construct the second equilateral triangle with a vertex on each side of the first triangle?[br][br]In geometry, a point can be defined by two lines (or segments or circles), where they cross. The point's location is determined by or located at the crossing of the two lines. However, a point cannot be defined by three lines -- that would be over-determining the point. Try to construct three lines (or segments or circles) to cross in one location and then use the point tool to place a point at that intersection. What happens?[br][br]Follow the hint. Analyze how things evolve as you drag point D along side AC. Describe what you see about dependencies and relationships among items in the figures.[br][br]Try to construct a pair of inscribed triangles that reproduce those dependencies or relationships.[br][br]Work together with your team-mates in your pod. This is a difficult challenge that usually takes people at least an hour to solve.[br][br]If you solve it, can you say why it works?

A "quadrilateral" is a four-sided figure. A pentagon has 5 sides. A hexagon has 6 sides. An octagon has 8 sides. [br]A "regular" quadrilateral has four sides of equal length and four angles of equal size (right angles). It is a square.[br]The slider in this challenge produces inscribed regular polygons of 3 to 9 sides. You can use the Regular Polygon tool (under the Polygon tool in the menu to create a regular polygon with a selected number of sides.[br][br]Can you construct an inscribed square? What did you notice by dragging point H and how did you use that in your construction.[br][br]Can you construct an inscribed regular pentagon? An inscribed regular hexagon? An inscribed regular octagon?

Work with your team-mates in your pod to complete the following proof that triangle DEF is equilateral:[br][br]Given an equilateral triangle ABC and points D, E, F on its sides such that AD = BE = CF, prove that inscribed triangle DEF is equilateral.[br]If AD = BE, then CD = AE because CD = AC - AD and AE = AB - BE; where AC = AB because they are equal sides of an equilateral triangle. Subtracting equal lengths from equal lengths leaves equal lengths.[br]....[br]Triangles ADE, BEF and CDF are congruent triangles because they have equal corresponding sides and included angles (SAS). Therefore corresponding sides DE = DF = DE, so the inscribed triangle DEF is equilateral, which is what was to be proven.