The host of Dancing with Triangles, Tom Twinkletoes, invites contestants on stage to perform their best dance with a triangle. The contestants dance with hexagons, squares, and even an elephant, but only one dances with a triangle. Tom explains why the other geometric shapes are not triangles.
The Earthlings need to split two apples and feed each Cerberus head an equal portion of apple. They use the Squak Pad to illustrate the equation to determine the solution and briefly explain the numerator and the denominator.
The Earthlings consult with Chief about whether or not Sleight O' Hand stole Hacker's recharger key. Chief claims that Sleight looks like a triangle and thus committed the crime, but Inez suggests that there could be other suspects. She states that they should look at the evidence from every point of view because shapes look different at different angles. They revisit the video footage to determine if Sleight O' Hand is still a suspect.
After determining which broom is the fastest, the CyberSquad needs to find out the speed of each broom in order to make it on TV. They measure the distance that each broom travels in five seconds and calculate the speed per second based on their measurements. Digit uses the Squak Pad to illustrate how they arrived to the solution.
The CyberSquad and Slider need to purchase an Alpha deck. When they hear the price, they calculate if they'll have enough snelfus to purchase the two remaining parts for the syncalator. Since they don't have time to wait for a used Alpha deck that is significantly cheaper, they purchase it and verify that they've received the correct amount of change from the salesperson.
The CyberSquad needs to get across a pool of lava. They find tiles in the shapes of triangles, pentagons, and hexagons, and tessellate the tiles in order to create a bridge that will shield them from the heat. When they run out of triangles in the middle of creating the bridge, they attempt to use pentagons, but that shape leaves gaps, so they use hexagons to complete the bridge.
The CyberSquad cannot make a big enough square with the rods it has available, so Jackie suggests making a triangle. The group successfully forms a triangle and then attempts to make a square with two triangles unsuccessfully. When Digit creates a right triangle, the group tries again to combine the triangles into a square. This time, it is successful, and Jackie explans how to know they've formed a proper square.
Grover and Rosita illustrate and explain the properties of a triangle. Rosita brings over two different-looking triangles to serve as visual aids, and Grover does not consider them to be triangles. He creates an equilateral triangle and states why it is a triangle. Rosita mentions that the triangles she brought in fit his definition of a triangle, causing Grover to realize that there are different types of triangles.