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Geometry 9-12 / Week 04 / Shop Day
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Week 04 · Proofs about Lines, Angles, Triangles and Parallelograms

Shop Day

Shop Day: the folding frame
// The parallelogram frame earns its name
⏱ about 20 min

Shop Day: The Folding Frame

Saturday the crew brings cardboard strips and paper fasteners to the kitchen table. "A frame that folds," Comet says.

Four strips, two long and two short, joined at the corners into a parallelogram. Comet pushes a corner and it leans.

"Still a parallelogram?" Wren asks Comet's cousin. "What do you notice?"

"The sides did not change, so the opposite sides are still equal. The angles changed."

Nova projects two strings across the corners. "Would you like a hint? Measure from each corner to the crossing."

The cousin measures. "Half and half. Both strings, every time we lean it."

"That is the theorem working," Comet says. "Your turn to run it."

  1. A proof is a chain of statements with reasons, from the given to the goal.
  2. Vertical angles, corresponding angles and alternate interior angles (across parallel lines) are congruent.
  3. Points on a perpendicular bisector are exactly those equally far from both ends.
  4. A triangle's angles add to 180°, base angles of an isosceles triangle match, a midsegment is half the third side.
  5. A parallelogram's opposite sides and angles are congruent and its diagonals bisect each other.
◇ BUILD A FOLDING PARALLELOGRAM
A grown-up cuts four cardboard strips with scissors: two of one length, two of another.
Join the strips corner to corner with paper fasteners so the frame can lean.
Stretch two strings corner to corner. Measure from each corner to the crossing. Compare the halves.
Lean the frame three ways and measure the four angles each time. Which pairs stay equal? Which pairs add to 180°?
For a grown-up
Cut the cardboard yourself with scissors and push the paper fasteners through, folding the legs flat.
Paper fasteners stay away from young children and pets.
If a measurement comes out unequal, ask what might have shifted. A loose fastener is a good lesson in itself.
SHOP DAY CHECK
  • Read the question.
  • Tap your answer.
A parallelogram ABCD with base 6 cm, height 3 cm and a 50 degree angle at AThe family frame leans to a 50° angle at one corner. What is the opposite angle?
A parallelogram ABCD with base 6 cm, height 3 cm and a 50 degree angle at AThe frame has a 50° angle at one corner. What is the angle at the next corner?
A triangle ABC with angles 50°, 65°, ?A cardboard brace for the frame has angles of 50° and 65°. What is its third angle?
One string across the family frame measures 90 cm. How far is it from a corner to the crossing, in centimeters? Type the number.
WHY THIS EXERCISEThe theorem predicts the measurement before anyone picks up the tape.
Draw the family frame leaning two different ways, with its diagonals and the crossing point marked on both.
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