Week 03 · Congruence from Rigid Motions
Shop Day
Shop Day: the family brace match
// Two braces match when a sequence of motions carries one onto the other
⏱ about 20 min
Shop Day: The Family Brace Match
Saturday the cardboard strips come home. Comet spreads them on the table: 6, 8 and 10, with holes at the ends.
"Family challenge," Wren says. "Each person builds a triangle from the same three strips. Then we compare."
Comet's cousin fastens his triangle and lays it on Comet's. "They match exactly."
"What do you notice?" Wren asks. "You never measured an angle."
"The sides forced the angles," the cousin says. "SSS."
Nova projects the SSA swing on the wall. "Would you like a hint? Now try two strips and an angle at the far end."
"Two triangles this time," Comet says. "Our designer, your family can test every criterion too."
- Two figures are congruent when a sequence of rigid motions carries one exactly onto the other.
- Congruent triangles have equal corresponding sides and angles, and the converse holds.
- SSS, SAS and ASA each force congruence. AAS works because the third angle is forced.
- SSA and AAA do not prove congruence: the swinging strip and the growing triangle are counterexamples.
- To check two braces, name the sequence of motions or match three parts in the right order.
◇ FAMILY BRACE MATCH
Together, cut cardboard strips 6, 8 and 10 centimeters long and punch a hole near each end.
Each person fastens the three strips into a triangle. Lay the triangles on each other. Do they all match?
Now take two strips and set a chosen angle between them. Measure the gap. Does everyone get the same gap?
Try the SSA swing: fix an angle at the far end and swing the short strip. Find both triangles and trace them.
Each person fastens the three strips into a triangle. Lay the triangles on each other. Do they all match?
Now take two strips and set a chosen angle between them. Measure the gap. Does everyone get the same gap?
Try the SSA swing: fix an angle at the far end and swing the short strip. Find both triangles and trace them.
For a grown-up
This week your student learned congruence as a sequence of rigid motions, and the criteria SSS, SAS and ASA.
Ask them why SSA fails. The answer should mention the swinging strip landing in two places.
You cut cardboard with scissors and push the paper fasteners through, folding the legs flat.
Ask them why SSA fails. The answer should mention the swinging strip landing in two places.
You cut cardboard with scissors and push the paper fasteners through, folding the legs flat.
Safety first
A grown-up cuts cardboard with scissors and pushes the paper fasteners. Students cut paper, sitting down.
Keep fastener points away from fingers and faces. A grown-up holds any ladder, and nothing heavy is lifted alone.
Keep fastener points away from fingers and faces. A grown-up holds any ladder, and nothing heavy is lifted alone.
SHOP DAY CHECK
- Read the question.
- Tap your answer.
Everyone built a triangle from the same three strips, and they all matched. Which criterion explains it?
The strip triangle has angles of about 37° and 53°. What is the third angle?A paper brace has corner A at (1, 1). Its twin is the brace slid 4 right. Where is A′?
How many different triangles did the SSA swing make? Type the number.
Count the places the swinging strip touched the ruler.
WHY THIS EXERCISETwo triangles from the same SSA parts is the counterexample that keeps SSA off the list.
Draw the family's SSS triangle and the two SSA triangles, each with its given parts marked.