Friday
Friday: Theorems Everywhere
Comet props the stepladder against the shop wall while a grown-up holds it. "Rungs. Parallel rails with a bunch of transversals."
"And the shelf brackets are triangles," Wren says. "Three angles, 180 every time. What do you notice about the folding table?"
Comet looks at the legs. "They cross. Vertical angles. And the whole leg frame is a parallelogram that stays parallel while it folds."
Nova projects a tiny grid over the table legs, the parallelogram glowing as it moves.
"Would you like a hint? The diagonals change, but the crossing stays at both midpoints."
"Because the theorem does not care how far it is folded," Comet says.
"Review today," Wren says. "Our designer, every theorem of the week, once more. Next week the model shrinks the whole set."
The week's theorems in everyday places
| Where | Theorem | Why it matters |
|---|---|---|
| the rungs of a ladder | corresponding angles across parallel rails | every rung meets both rails at the same angle, so it sits level |
| a folding table's leg frame | a parallelogram's opposite sides stay congruent | the top stays level as the legs swing |
| a triangular shelf bracket | the angles add to 180° | two measured angles give the third without a protractor |
| a roof gable with equal slopes | base angles of an isosceles triangle | equal sides force equal angles at the eaves |
| a picture frame checked with two strings | a rectangle's diagonals are congruent | equal diagonals mean the corners are square |
The medians meet at one point
A median joins a corner of a triangle to the midpoint of the opposite side. Draw all three on a cardboard brace.
They cross at one point, and the brace balances on a pencil tip there. The proof uses midsegments: each median cuts the others two to one.
You will meet the full proof again with coordinates in week 9.
The week in five lines
- A proof is a chain of statements with reasons: given, definitions, postulates, earlier theorems.
- Vertical angles are congruent. Across parallel lines, corresponding and alternate interior angles are congruent.
- The points on a perpendicular bisector are exactly the points equally far from both ends.
- A triangle's angles add to 180°, an isosceles triangle's base angles match, and a midsegment is half the third side.
- A parallelogram's opposite sides and angles are congruent and its diagonals bisect each other; a rectangle's diagonals are congruent.
- Read the question.
- Tap your answer.
On the ladder, a rung meets the left rail at 70° (angle a). What is angle g at the other rail?
A shelf bracket has angles of 55° and 55°. What is the third angle?
A gable is an isosceles triangle with a 100° peak. What is each angle at the eaves?
A folding table's leg frame has a 115° angle. What is the angle next to it?- Read the question.
- Tap your answer.
| Week review | True or false? |
|---|---|
| Vertical angles are congruent. | ? |
| Alternate interior angles are congruent on any two lines, parallel or not. | ? |
| 40 + 70 + 70 = 180 | ? |
| A midsegment of a triangle is twice the third side. | ? |
| The diagonals of every parallelogram bisect each other. | ? |
| The diagonals of every parallelogram are congruent. | ? |
Draw the three medians of a paper triangle and balance it on a pencil tip where they cross.
Sharp review, designer. Tomorrow is Shop Day: a folding parallelogram for the family to build and test.