Congruence & Sequences of Transformations — Print

Two shapes are congruent exactly when some sequence of slides, flips and turns carries one onto the other.

What the practice questions cover

These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Two pieces fall out of the same cookie cutter. One lands flat, the other lands turned.

Are they the same piece? You cannot just look — turning makes shapes look different.

two pieces cut from the same die piece A piece B, turned turn it one tick fits one tick, two fit two, three fit three

So mark the sides. One tick on the short side, two on the medium, three on the long.

Now the question answers itself: one tick fits one tick, two fit two, three fit three.

Turning the piece never changed how long its sides were, so the ticks still match.

Congruent does not mean "looks the same." It means some sequence of slides, flips and turns carries one exactly onto the other — and the tick marks are how you check without doing the carrying.

Congruent triangles with matching sides marked congruent = same size and same shape, however it has been moved A B C A' B' C' rotate one tick matches one tick: AB = A'B', AC = A'C', BC = B'C'

A definition you can actually use

Older books define congruent as "same size and shape". True, but it does not tell you what to do.

The useful version: two figures are congruent if a sequence of rigid motions maps one exactly onto the other.

So to prove congruence, describe the moves. To use congruence, read off equal sides and equal angles.

Order of the letters matters

Writing △ ABC ≅ △ DEF claims A matches D, B matches E, C matches F. Get the order right and every equal pair falls out for free.

Reading corresponding parts

From △ ABC ≅ △ DEF you may immediately write AB = DE, BC = EF, AC = DF, and ∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F.

That is six facts from one statement. Most congruence questions are just asking you to pick the right one.

Line the letters up

Write ABC above DEF on two lines. Now the matching letters are in the same column, and you can read off pairs without thinking.

Sequences: doing two moves

Do the first move completely, write down the new coordinates, then do the second move on those.

Reflect across the y-axis, then translate 2 up: first (x, y) → (-x, y), then that result goes to (-x, y + 2).

Order usually matters. Reflect-then-translate and translate-then-reflect often land in different places.

Two moves in sequence: reflect triangle ABC in the y-axis, write down the new point A′(−1, 1), then translate 2 up to A″(−1, 3) finish the first move, write the new point down, then do the second xymirror ABC A′A″ step 1 · reflect in the y-axis (x, y) → (−x, y) A(1, 1) → A′(−1, 1) step 2 · translate 2 up (−x, y) → (−x, y + 2) A′(−1, 1) → A″(−1, 3) after step 1 — write it down after step 2

Never merge the steps in your head

Write the intermediate point down. Combining two rules mentally is the single biggest source of errors in this unit.

Orientation tells you how many flips

Translations and rotations keep the orientation: if A, B, C ran clockwise, so do A', B', C'.

A reflection reverses it. Two reflections reverse it twice, so back to the original.

If the image is flipped over, an odd number of reflections happened. That narrows down your answer fast.

Orientation: a rotation keeps A, B, C running clockwise, one reflection reverses it, a second reflection restores it follow A → B → C round: a reflection reverses the direction, a turn never does originalABCclockwiserotatedA′B′C′still clockwisereflectedA′B′C′anticlockwisereflected twiceA″B″C″clockwise again flipped over? Then an odd number of reflections happened

Worked examples

Reading off a congruence statement

Given △ PQR ≅ △ STU with PQ = 9 and ∠ U = 40°, find ST and ∠ R.

  1. Line them up: P→S, Q→T, R→U. The order of the letters is the correspondence.
  2. PQ corresponds to ST, so ST = 9. Corresponding sides of congruent triangles are equal.
  3. ∠ R corresponds to ∠ U, so ∠ R = 40°. Corresponding angles are equal too.

ST = 9 and ∠ R = 40°

Carrying out a sequence

Point A(3, -1) is reflected across the x-axis, then translated 4 left. Where does it end up?

  1. Reflection across the x-axis: (3, -1) → (3, 1) The y changes sign.
  2. Now translate that point 4 left: (3, 1) → (-1, 1) Subtract 4 from the x of the new point, not the old one.

(-1, 1)

Describing the move

Triangle ABC has A(1, 1), B(4, 1), C(1, 3). Its image has A'(1, -1), B'(4, -1), C'(1, -3). What single transformation was used?

  1. Compare x values: 1 → 1, 4 → 4, 1 → 1. Unchanged. So nothing moved left or right.
  2. Compare y values: 1 → -1, 1 → -1, 3 → -3. Each one changed sign. That is exactly the rule (x, y) → (x, -y).
  3. So it is a reflection across the x-axis. Check orientation: it is reversed, which fits a single reflection.

A reflection across the x-axis

Words to know

Congruent
Same size and same shape. Written △ ABC ≅ △ DEF.
Corresponding parts
The pieces that match up: A with D, side AB with side DE, and so on.
Sequence of transformations
Two or more moves done one after another.
Orientation
Whether the labels run clockwise or counter-clockwise around the shape.

Mistakes to watch for

Writing the letters in the wrong order. △ ABC ≅ △ DEF is a claim that A matches D, B matches E and C matches F. Write △ DFE instead and you have claimed six different facts, most of them false.

Applying the second transformation to the original point. In a sequence, step two acts on the result of step one. Write the middle point down on its own line — it is the single cheapest way to stop this error.

Assuming order does not matter. Reflect-then-translate and translate-then-reflect usually land in different places, because the mirror does not move with the shape. Try (1, 1) both ways and you will see them separate.

Calling two shapes congruent because they look alike. Two rectangles can look similar and have quite different side lengths. Congruence is a measurement claim, so it needs matching measurements, not a glance.

The symbol ≅ is an equals sign with a ∼ on top. The ∼ means "same shape" and the = means "same size."

So ∼ alone is similar and ≅ is congruent — the notation is telling you that congruent is similar plus one extra condition.

It was introduced by the German mathematician Gottfried Leibniz in the 1670s, the same man who invented much of the notation used in calculus.

Problem 1

△ ABC ≅ △ DEF and AB = 7. Find DE.

Worked answer

7

AB corresponds to DE

Corresponding sides are equal

DE = 7

Problem 2

△ PQR ≅ △ XYZ and ∠ Q = 55°. Find ∠ Y.

Worked answer

55

∠ Q corresponds to ∠ Y

∠ Y = 55°

Problem 3

△ LMN ≅ △ RST with MN = 12. Which side of △ RST is 12?

Worked answer

ST

MN corresponds to ST

ST = 12

Problem 4

A square is rotated 45°. What is its new side length if the original was 6 cm?

Worked answer

6

Rotation changes position, not size

The side is still 6 cm

Problem 5

Which sequence guarantees the image is congruent to the original?

  1. Reflect, then translate
  2. Dilate by 2, then translate
  3. Reflect, then dilate by 3
  4. Dilate by 12, then rotate
Worked answer

Reflect, then translate

Reflections and translations are both rigid

Dilations are not, unless k = 1

So the first sequence is the only one

Problem 6

A(4, 1) is reflected across the x-axis and then translated 3 left. Find the image.

Worked answer

(1, -1)

Reflect: (4, 1) → (4, -1)

Translate 3 left: (1, -1)

Problem 7

B(-2, 5) is translated 6 right and then reflected across the y-axis. Find the image.

Worked answer

(-4, 5)

Translate: (-2, 5) → (4, 5)

Reflect across the y-axis: (-4, 5)

Problem 8

△ ABC ≅ △ DEF with AB = 3x - 4 and DE = 2x + 5. Find x.

Worked answer

9

3x - 4 = 2x + 5

x = 9

Check: 3(9) - 4 = 23 and 2(9) + 5 = 23 ✓

Problem 9

△ GHI ≅ △ JKL with ∠ G = (4y + 10)° and ∠ J = (6y - 14)°. Find y.

Worked answer

12

4y + 10 = 6y - 14

24 = 2y

y = 12

Problem 10

A shape is reflected twice, across two different lines. Is the final orientation the same as the original or reversed?

Worked answer

same

One reflection reverses it

A second reflection reverses it back

So the orientation is the same

Problem 11

A figure and its image have opposite orientations. Which sequence could have produced it?

  1. A reflection followed by a translation
  2. Two rotations
  3. A translation followed by a rotation
  4. Three rotations
Worked answer

A reflection followed by a translation

Rotations and translations keep the orientation

An odd number of reflections reverses it

So the answer must contain exactly one reflection

Problem 12

△ ABC ≅ △ DEF. The perimeter of △ ABC is 31. Find the perimeter of △ DEF.

Worked answer

31

Every corresponding side is equal

So the perimeters are equal

31

Problem 13

P(1, 4) is rotated 180° about the origin and then translated by (x + 5, y - 2). Find the image.

Worked answer

(4, -6)

Rotate: (1, 4) → (-1, -4)

Translate: (-1 + 5, -4 - 2) = (4, -6)

Problem 14

△ ABC ≅ △ DEF. AB = 5, BC = 8 and the perimeter of △ DEF is 20. Find AC.

Worked answer

7

Perimeter of △ ABC is also 20

5 + 8 + AC = 20

AC = 7

Problem 15

Which single transformation has the same effect as reflecting across the x-axis and then across the y-axis?

Worked answer

a 180 degree rotation about the origin

(3, 5) → (3, -5) → (-3, -5)

Both coordinates ended up negated

That is the rule for a 180° rotation about the origin

Problem 16

△ ABC ≅ △ DEF, ∠ A = 40° and ∠ E = 75°. Find ∠ C.

Worked answer

65

∠ B = ∠ E = 75°

∠ C = 180 - 40 - 75

∠ C = 65°

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