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Geometry 9-12 / Week 03 / Tuesday
2/6
Week 03 · Congruence from Rigid Motions

Tuesday

Three parts are enough
// Two braces match when a sequence of motions carries one onto the other
⏱ about 20 min

Tuesday: Three Parts Are Enough

Wren lays two braces on the bench. "I measured three parts of each and they match. Are the braces congruent?"

"Which three?" Comet asks. "Three sides, I believe. Three angles, I do not."

"Two sides and the angle between them," Wren says. "What do you notice if I line up that angle?"

Comet slides one brace so the matching corners meet, then turns it so one side lies along the other.

"The angle fixes the direction of the second side. The length fixes where it ends. The third corner is forced."

Nova projects the third sides. They lie exactly on each other. "Would you like a hint? That is SAS."

"Side, angle, side. One sequence, no guessing," Wren says. "Our designer, find the motion for each case."

The three criteria

CriterionMatching parts, in order around the triangleWhy it forces congruence
SSSthree sidesthree set lengths can close in only one shape
SAStwo sides and the angle between themthe angle sets the direction, the lengths set the endpoints
ASAtwo angles and the side between themthe side sets two corners, the angles set the two rays that meet at the third

Each criterion names a sequence of rigid motions that must land one triangle on the other.

AAS also works, because the third angle is forced once two are known. Then it is ASA in disguise.

Two braces side by side with two sides and the included angle marked as equal: the SAS case.

Why SAS follows from rigid motions

  1. Slide triangle 1 so its marked corner lands on the marked corner of triangle 2.
  2. Turn triangle 1 about that corner until the first marked side lies along the matching side of triangle 2.
  3. If the third corner is on the wrong side, reflect triangle 1 over that shared side.
  4. The equal angle makes the second sides point the same way. The equal lengths make them end at the same point.
  5. All three corners now coincide, so the triangles are congruent.

Notice what SAS used: a translation, a rotation and maybe a reflection. Nothing else was needed.

ASA runs the same way. After the side lines up, the two equal angles aim two rays that can meet at only one point.

Two braces side by side with two angles and the side between them marked as equal: the ASA case.

Two methods for one check

To show two braces are congruent, name the whole sequence of motions, or match three parts with SSS, SAS or ASA.

The sequence is the definition. The criterion is the shortcut. The shortcut works only because the sequence always exists.

WHICH CRITERION?
  • Read the question.
  • Tap your answer.
Two braces match in two sides and the angle between them. Which congruence criterion applies?
Two braces match in all three sides. Which congruence criterion applies?
Two braces match in two angles and the side between them. Which congruence criterion applies?
THE CORNER BRACE
  • Read the question.
  • Tap your answer.
A triangle ABC with angles 30°, 60°, ?A corner brace has angles of 30° and 60°. What is its third angle?
Brace ABC is congruent to brace DEF, A to D, B to E, C to F. AB is 100 cm. How long is DE?
Brace ABC is congruent to brace DEF, A to D, B to E, C to F. BC is 50 cm. How long is EF?
SAS, STEP BY STEP
  • ?Slide triangle 1 so the marked corner lands on the matching corner
  • ?Turn triangle 1 so the first marked side lies along the matching side
  • ?Reflect over the shared side if the third corner is on the wrong side
  • ?The equal angle and equal length force the second side to coincide
  • ?All three corners coincide, so the triangles are congruent
WHY THIS EXERCISEThis order is the proof. Each motion is rigid, so nothing changes along the way.
Two sides and the angle between them match. Type the three letters of the criterion.
The angle between two given sides is called this kind of angle. Type the one word.
Try it
Cut two paper triangles with the same two sides and the same angle between them. Use a ruler and a protractor.
Slide, turn and flip one onto the other. Write the sequence you used.
Draw two triangles that match by ASA. Mark the two equal angles with arcs and the equal side with one tick.

Excellent reasoning. Tomorrow is Shop Lab: cardboard strips and paper fasteners test every criterion.

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