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

Thursday

The brace data
// Two braces match when a sequence of motions carries one onto the other
⏱ about 20 min

Thursday: The Brace Data

Wren pins a page to the corkboard: four pairs of braces, each with the parts the crew measured. "Sort them."

"Congruent or not, and why," Comet says. "Pair one: three sides each, all matching. SSS. Done."

"Pair two: all three angles match but the sides are different sizes," Wren says. "What do you notice?"

"Same shape, different size. AAA. Not congruent."

Nova projects pair three: two sides and the right angle between them. "Would you like a hint? Which side is forced?"

"The third one," Comet says. "SAS, congruent, and I can compute the third side."

"Pair four is the SSA swing," Wren says. "Two different braces. Our designer, check every row of our table."

The crew's brace table

Every number here is the crew's own made-up Scene Shop measurement in centimeters and degrees. Nothing is a real-world fact.

PairBrace 1Brace 2Matching partsVerdict
1sides 50, 86.6, 100sides 50, 86.6, 100S, S, Scongruent (SSS)
2angles 40°, 60°, 80°, sides 52.2, 70.4, 80angles 40°, 60°, 80°, sides 78.3, 105.5, 120A, A, Anot congruent
3sides 60, 80, angle 90° betweensides 60, 80, angle 90° betweenS, A, Scongruent (SAS)
4angle 30°, sides 6, 10, third side 12angle 30°, sides 6, 10, third side 5.3S, S, Anot congruent

Pair 3: both braces have sides 60 and 80 around a right angle. SAS forces the third side, and the crew measured 100 on both.

Pair 1: both braces have the angles 30°, 60°, 90° as well, because SSS forces every angle too.

Pair 2: the braces have the same angles but sides 52.2, 70.4, 80 against 78.3, 105.5, 120. Same shape, not congruent.

Two braces with the same three angles of 40, 60 and 80 degrees but different side lengths: the AAA case.
The SAS brace with sides 60 and 80 centimeters around a right angle and the forced third side labeled.
SORT THE PAIRS
  • Read the question.
  • Tap your answer.
Pair 1: both braces have sides 50, 86.6 and 100. Which criterion proves them congruent?
Pair 2: both braces have angles 40°, 60° and 80°, and nothing else is known. What does this prove?
Pair 3: both braces have sides 60 and 80 with a 90° angle between them. Which criterion proves them congruent?
USE THE TABLE
  • Read the question.
  • Tap your answer.
Pair 1: brace ABC is congruent to brace DEF, A to D, B to E, C to F. CA is 86.6 cm. How long is FD?
Pair 1: brace 1 has sides 50, 86.6, 100 cm. What is the perimeter of its congruent twin, in cm?
A triangle ABC with angles 40°, 60°, ?Pair 2: two angles of each brace are 40° and 60°. What is the third angle?
Pair 3 matches in two sides and the angle between them. Type the three letters of the criterion.
WHY THIS EXERCISENaming the criterion is how the crew records that the braces will fit the same frame.
StatementTrue or false?
Pair 1 is congruent by SSS.?
Pair 2 is congruent because all three angles match.?
In pair 3, the third side of each brace must be the same length.?
Pair 4 shows that SSA does not prove congruence.?
Congruent braces always have the same perimeter.?
WHY THIS EXERCISESorting pairs by criterion is exactly the job the braces need before they go into the frame.
Try it
On paper, draw a triangle with sides 6, 8 and 10 centimeters. Draw a second one with the same sides in a different order.
Cut both out and check: can you slide, flip and turn one onto the other?
Draw pair 3: two right triangles with legs 60 and 80 (scaled down), marked SAS, with the third side labeled.

Sharp sorting, designer. Tomorrow you spot congruence in everyday life and review the week.

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