Slipping rods and collapsing triangles

 Activity 1

Support a two-legged structure hinged at $C$, with legs $CA$ and $CB$. If we press $C$ downwards, supporting the structure vertically; $A$ and $B$ taking new positions $A'$ and $B'$. What happens to angles at new positions of $A$, $B$ and $C$? Bring $C$ further down, and observe the changes at the new positions of $A$, $B$, $C$. Press $C$ further downwards, almost reaching the floor level. The angles at $A$ and $B$ now approach zero. What approaches the angle at $C$ to? What seems to be the sum of the three angles at $A$, $B$ and $C$ at this stage?

Slipping Two-Legged Structure

Slowly, raise the level of $C$, repeating the experiment in the reverse order. Did you keep observing the changes in the angles at $A$, $B$ and $C$? What looked happening to the sum of the angles?
Repeat the experiment with the view to answer the questions better.

Let us repeat the experiment with a structure with $AC$ and $CB$ of different lengths. Again observe what increase of angle at $C$ causes for angles at $A$ and $B$. Next make a measured increase in the angle at $C$, and observe the changes at $A$ and $B$. And what happens to the sum of
the angles?

 Shall we take up the reverse activity? Let $C$ be raised higher and  higher; and yet higher.  The angle at $C$ becomes smaller and angles  at $A$ and $B$ become larger. What about the sum total of the angles?

 Activity 2

 Let a rod $AB$ rest with end $A$ on the wall and the end $B$ at the  floor. I hope it is so placed that the foot of the perpendicular from  $ A$ to the floor and foot of the perpendicular from $B$ to the wall
 coincide at $C$, and angle $ACB$ is ninety degrees. If the rod slips, $A$ taking positions $A'$, $A''$ .... on the wall and $B$ taking positions $B'$, $B''$ ... on the floor; observe that angle at new  positions of $A$ increases, while angles at new positions of $B$  diminishes.

Slipping rods

When the rod is almost horizontal, the $A$-end of the rod  is nearly on $C$; and the $B$-end makes almost zero angle to the  floor and the angle at $A$ is nearly ninety degrees. What is the sum of the angles at $A$, $B$ and $C$ in different positions of the rod? Repeat the experiment, raising the rod straight; with $A$ going up  the wall and $B$ nearing $C$. Observe the angles; when $B$ is almost at $C$. What about the sum of the angles at $A$, $B$ and $C$? Shall  we try to formulate our observations about the sum of the angles of a  triangle.