Informal Transformational Proof Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Informal Transformational Proof.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An informal transformational proof uses translations, rotations, reflections, and dilations to explain why two figures are congruent or similar.

Instead of measuring sides and angles, show that one shape can be moved, flipped, or resized to land exactly on another.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A transformational proof explains why figures match by describing transformations that carry one to the other.

Common stuck point: The procedure for informal transformational proof is the easy part; the trap is listing transformations without saying what they preserve. Asking "Can I describe transformations that carry one figure onto the other?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I describe transformations that carry one figure onto the other?

Worked Examples

Example 1

medium
Prove informally that the diagonals of a rectangle are congruent using a reflection.

Answer

Reflect across the perpendicular bisector of a side; this swaps the two diagonals, so they have equal length

First step

1
A rectangle has a line of symmetry through the midpoints of opposite sides.

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Example 2

medium
Prove informally that opposite sides of a parallelogram are congruent using a 180° rotation about its center.

Example 3

medium
Use a 180° rotation about the midpoint of a side to prove the triangle midsegment is half the third side.

Example 4

hard
Prove informally that any two congruent segments can be carried onto each other by a single rigid motion.

Example 5

hard
Two parallel lines are cut by a transversal. Use a translation along the transversal to argue corresponding angles are congruent.

Example 6

challenge
Use a sequence of two reflections to map segment ABAB at angle 30° to segment A′B′A'B' at angle 90° (same endpoint A=A′A=A'). Describe the reflection axes.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which transformation preserves both size AND orientation: rotation, reflection, or dilation by 1?

Example 2

easy
Triangle A is translated 3 right and 2 up to triangle B. Are A and B congruent?

Example 3

easy
Reflection across the xx-axis sends (a,b)(a,b) to which point?

Example 4

easy
Rotation by 90° about the origin sends (1,0)(1,0) to which point?

Example 5

medium
If you translate, then reflect, can the result be done by a single transformation?

Example 6

hard
Triangle AA vertices (0,0),(2,0),(0,2)(0,0),(2,0),(0,2); triangle BB vertices (0,0),(0,2),(−2,0)(0,0),(0,2),(-2,0). Single transformation from AA to BB?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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