← Back to course
Geometry 9-12 / Week 03 / Monday
1/6
Week 03 · Congruence from Rigid Motions

Monday

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

Monday: Two Braces, One Motion

Comet holds up two triangular braces, one in each hand. "Same brace? They look the same. How do we know?"

"Last week's test," Wren says. "If a sequence of rigid motions carries one onto the other, they are congruent."

Comet lays one brace on the chalk grid, flips it over the y-axis, then slides it 5 down. It covers the other exactly.

"Every corner landed," she says. "Every side, every angle."

Nova projects the two outlines, now one. "Would you like a hint? Rigid motions keep lengths. So what must be equal?"

"All three sides and all three angles," Wren says. "Six matching parts, from one sequence."

"Our designer," Comet says, "predict where every corner lands before we move the next brace."

Comet and Wren hold up two matching triangular braces in the Scene Shop while Nova projects their outlines.

Congruent means a motion carries one onto the other

Two figures are congruent when a sequence of rigid motions carries one exactly onto the other.

The parts that land on each other are corresponding parts. Rigid motions keep every distance and angle, so corresponding parts are equal.

The other way works too. If two triangles have three equal sides and three equal angles, a sequence of motions carries one onto the other.

A solved problem to study

Wren's brace has corners (1, 1), (5, 1), (2, 4). The twin has corners (-1, -4), (-5, -4), (-2, -1). Here is his work.

Reflect over the y-axis: (1, 1), (5, 1), (2, 4) go to (-1, 1), (-5, 1), (-2, 4). Then slide 5 down: they land on (-1, -4), (-5, -4), (-2, -1).

Every corner of the brace lands on a corner of the twin. So the braces are congruent, and side AB (4 units) equals side A′B′ (4 units).

Why this is enough: the sequence is made of rigid motions, and rigid motions change no length and no angle.

The brace in gold and its congruent twin in blue, after a reflection over the y-axis and a slide 5 down.
Two congruent corner braces side by side, each with angles 30, 60 and 90 degrees and matching tick marks on equal sides.

The crew's corner braces measure 50, 86.6, 100 centimeters, with angles 30°, 60°, 90°. Matching tick marks show which parts correspond.

PREDICT WHERE THE CORNER LANDS
  • Read the question.
  • Tap your answer.
Corner C of the brace is at (2, 4). The brace reflects over the y-axis. Where does C′ land?
After the reflection, corner B′ is at (-5, 1). The brace slides 5 down. Where does it land?
Corner C at (2, 4) turns 90° counterclockwise about the origin. Where does C′ land?
CONGRUENT OR NOT?
  • Read the question.
  • Tap your answer.
Triangle 1 has corners (1, 1), (5, 1), (2, 4). Triangle 2 has corners (-1, -4), (-5, -4), (-2, -1). Are they congruent?
A grid with a figure at (1, 1), (5, 1), (2, 4) and its image in blueTriangle 1 has corners (1, 1), (5, 1), (2, 4). Triangle 2 has corners (1, 1), (6, 1), (2, 4). Are they congruent?
Triangle 1 has corners (1, 1), (5, 1), (2, 4). Triangle 2 has corners (-1, 1), (-1, 5), (-4, 2). Are they congruent?
StatementTrue or false?
Congruent figures are carried onto each other by a sequence of rigid motions.?
A stretch can be one of the motions in a congruence sequence.?
Corresponding sides of congruent triangles are equal.?
Two triangles with all six parts equal might still not be congruent.?
WHY THIS EXERCISEThe definition of congruence is the one tool every proof in this course will reach for.
Side AB of the brace is 4 units long. How long is its corresponding side A′B′ on the twin? Type the number.
WHY THIS EXERCISEOne matching side is the first of six matching parts that congruence guarantees.
Try it
On graph paper, draw the brace at (1, 1), (5, 1), (2, 4). Cut a tracing of it.
Flip the tracing over the y-axis and slide it 5 down. Does it land on (-1, -4), (-5, -4), (-2, -1)?
Safety first
Cut paper with scissors, sitting down. A grown-up cuts cardboard braces and holds any ladder.
Draw the brace (1, 1), (5, 1), (2, 4) and its twin (-1, -4), (-5, -4), (-2, -1). Mark the corresponding corners with matching colors.

Strong start, designer. Tomorrow you learn why three parts, in the right order, are enough.