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Geometry 9-12 / Week 03 / Friday
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Week 03 · Congruence from Rigid Motions

Friday

Congruence everywhere
// Two braces match when a sequence of motions carries one onto the other
⏱ about 20 min

Friday: Congruence Everywhere

Comet stacks the finished braces. "Every one has to be congruent, or the frame will lean."

"Like a shelf bracket and its twin," Wren says. "Or the two halves of a folded paper heart. What do you notice about those?"

"A reflection carries one half onto the other," Comet says. "Fold, and they match."

Nova projects a tile floor, every tile the same. "Would you like a hint? Which motion carries one tile onto its neighbor?"

"A translation. And the braces matched by a flip and a slide."

"Review day," Wren says. "Our designer, every criterion and every counterexample. Next week we prove things about lines and angles."

Congruence in everyday places

WhereWhy congruentMotion
two halves of a folded paper shapethe fold is a line of reflectionreflection
a row of identical floor tileseach tile slides onto the nexttranslation
matching brackets on a shelfmade to the same three sidesSSS
the two braces in a frame cornersame two sides around the same right angleSAS

In each case, a sequence of rigid motions carries one copy onto the other. That is the definition at work.

The brace in gold and its congruent twin in blue after a 90 degree rotation about the origin.

The week in five lines

  • Congruent: a sequence of rigid motions carries one figure exactly onto the other.
  • Congruent triangles have three pairs of equal sides and three pairs of equal angles, and the converse holds.
  • SSS, SAS and ASA each force a sequence of rigid motions, so each proves congruence. AAS works through the angle sum.
  • SSA and AAA do not: the swinging strip gives two triangles, and equal angles allow any size.
  • To check two braces, name the sequence or match three parts in the right order.
SPOT THE MOTION
  • Read the question.
  • Tap your answer.
Two halves of a folded paper shape match exactly. Which rigid motion carries one onto the other?
A row of identical tiles runs along the shop wall. Which rigid motion carries one tile onto the next?
A grid with a figure at (1, 1), (5, 1), (2, 4) and its image in blue and a center at (0, 0)The brace has corners (1, 1), (5, 1), (2, 4). Its twin has corners (-1, -1), (-5, -1), (-2, -4). Which sequence carries it onto the twin?
WEEK REVIEW
  • Read the question.
  • Tap your answer.
Two triangles match in these parts, in order around the triangle: angle, side, angle. Which congruence criterion applies?
Two triangles match in these parts, in order around the triangle: side, side, angle. Which congruence criterion applies?
A triangle ABC with angles 50°, 60°, ?A brace has angles of 50° and 60°. What is its third angle?
Brace 1 has corners (1, 1), (5, 1), (2, 4). Brace 2 has corners (1, 1), (6, 1), (2, 4). Are they congruent?
Week reviewTrue or false?
A sequence of rigid motions carrying one triangle onto another makes them congruent.?
SSA proves two triangles congruent.?
Two triangles with the same three angles must be the same size.?
Corresponding angles of congruent triangles are equal.?
SAS needs the angle between the two given sides.?
WHY THIS EXERCISEThese five facts carry every proof in the next weeks, starting with angles and parallel lines.
OUR WEEK, IN ORDER
  • ?We defined congruent by a sequence of rigid motions and predicted where every corner lands.
  • ?We learned SSS, SAS and ASA and why each follows from rigid motions.
  • ?We fastened cardboard strips in Shop Lab and found that SSA and AAA leave a choice.
  • ?We sorted the crew's brace pairs by criterion.
WHY THIS EXERCISEEach day added one more reason the shortcut criteria are safe to use.
Try it
Fold a sheet of paper and cut a shape through both layers. Unfold it and name the motion that matches the halves.
Find two identical objects at home and write which rigid motion carries one onto the other.
Draw two congruent triangles that match by SAS. Mark the equal sides with ticks and the included angle with an arc.

Terrific week. You can define, test and prove congruence. Tomorrow is Shop Day.

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