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MATH · GRADE 7Geometry

Congruent Polygons

△ABC ≅ △RST is six claims in five symbols. Can you read all six?

Grade 7
△ABC ≅ △RSTABCTRS?six claims, unread…
In this lesson

What congruent polygons are

Two polygons are congruent if you can move one onto the other so they match exactly: same side lengths, same angles, same shape, same size. The notation is ≅.

A congruence statement names more than just the fact. It names the correspondence. When we write

△ABC≅△RST\triangle ABC \cong \triangle RST

the letter order forces a specific vertex pairing:

  • A corresponds to R (first letter to first letter),
  • B corresponds to S (second to second),
  • C corresponds to T (third to third).

From that pairing, every side and every angle has a partner: A̅B̅ ≅ R̅S̅, B̅C̅ ≅ S̅T̅, C̅A̅ ≅ T̅R̅; ∠A ≅ ∠R, ∠B ≅ ∠S, ∠C ≅ ∠T. Six congruences, all packed into the one notation.

The same rule extends to four-vertex polygons. ABCD ≅ PQRS forces A↔P, B↔Q, C↔R, D↔S: four matched pairs of sides and four matched pairs of angles. The letter order is what carries that information.

See it: match the vertices

The widget below shows two polygons side by side. Tap a vertex of the left polygon, then tap the matching vertex on the right. When all vertices are paired, the widget checks whether the chosen correspondence is geometrically valid, and if it is, displays the correct congruence notation in △…≅△… (or ABCD ≅ …) form.

Match the vertices

Tap each vertex of the second polygon to pair it with the next vertex of the first. When all are paired, the widget checks side and angle equalities.

Two polygons appear side by side. The student taps a vertex on the left polygon, then taps the vertex it matches on the right polygon. Once all pairs are made, the widget verifies the correspondence (comparing side lengths and angles) and either displays the correct congruence notation or flags the mismatch. Presets cycle through triangles in standard orientation, triangles where the second is rotated (so matching by position not by letter alphabet is required), and a four-vertex polygon for the ABCD ≅ RSTU case.
Preset:
ABCRST

Order matters in the notation: the i-th letter on the left maps to the i-th letter on the right.

A few patterns to notice as you work the widget:

  • Order is the matching. Once you've paired A↔R, B↔S, C↔T, the notation △ABC ≅ △RST writes itself. Reverse the matching and the notation changes too.
  • The rotated preset is the test. When the second triangle's vertices aren't labelled in alphabetical order, matching by position, not by letter, is the only thing that works.
  • Four vertices is the same rule. ABCD ≅ PQRS is a four-pair matching. Nothing about the rule changes when the polygon has more sides.

Reading the marks on a diagram

Most geometry problems don't say "AB is congruent to DE" in words. They draw the figure with hash marks on segments and arcs inside angles. Same number of marks = congruent group. One hash matches one hash; two hashes match two hashes; never one to two.

Read the marks

Tap any two marked elements. If they share the same marking (same number of hashes or arcs), they're congruent.

A figure with marked segments (hash marks) and marked angles (arcs). The student taps any two marked elements and the widget reports whether they share the same marking. Same number of marks → congruent; different number → not congruent (different mark groups). Presets include a triangle pair with single hashes and single arcs, and a quadrilateral pair with mixed mark counts on different sides.
Diagram:
ABCDEF
Tap a marked segment or angle to start.

Same number of marks = congruent group. Different number = different group.

The widget reports the congruence in standard notation. When you read a diagram in an exam or a textbook, do the same scan on your own: look at every segment for hash marks, every angle for arcs, and write down the implied congruences before solving.

Where it shows up in real life

The Drumheller Hoodoos, just east of the Red Deer River along Highway 10X, are sandstone columns shaped by wind and water erosion. Several of them stand close together in clusters, and tourist photographs often show two hoodoos that look like mirror images of each other: same height, same cap-rock width, same narrow waist. That visual symmetry is congruence in the wild: if two hoodoos really are congruent, every measurement on the second matches the corresponding measurement on the first.

Rose windows on Alberta churches and community halls do the same thing more deliberately. The petals or quadrilateral panes around the rim are designed to be congruent: same shape, same size, rotated copies of one master pattern. Builders use the correspondence directly: cut one pane, then trace it for the others so the angles and sides all match.

Worksheet

These aren't graded. Get them right, get them wrong. The goal is to feel out where the correspondence lands.

Practice · Not graded

MA.7.GEO.1

Practice the idea

01 / 09

Given △ABC ≅ △RST, which list shows the correct corresponding sides and angles?

Multiple choice: given △ABC ≅ △RST, identify the corresponding sides and angles.
Show common mistakes

Student says

“Says two triangles are congruent without checking all corresponding parts.”

What it reveals

Treats 'looks like' as 'is.' Hasn't internalised that congruence requires VERIFICATION of all corresponding sides and angles.

Targeted response

In the Congruence Mapper above, drag through a few presets and watch the verification step: every side and every angle is checked. Congruence requires ALL corresponding parts equal. Visual similarity isn't enough; two triangles can look similar in shape but differ in size (those are 'similar', not 'congruent').

Student says

“Writes △ABC ≅ △XYZ when the actual correspondence is A↔X, B↔Z, C↔Y.”

What it reveals

Knows the triangles are congruent but didn't write the letter order correctly. The letter order specifies the correspondence.

Targeted response

Order matters. If the actual correspondence is A↔X, B↔Z, C↔Y, the correct notation is △ABC ≅ △XZY. The order of letters in the second triangle must reflect the matching with the first triangle's letters in order. Read off the matching, then write the notation in that order.

Student says

“Misses hash marks or arc symbols on a diagram and tries to solve without using the marked congruences.”

What it reveals

Diagram literacy gap. Markings encode information the problem doesn't repeat in words.

Targeted response

Use the Marking Reader above as a discipline. Before solving any geometry problem, scan every segment for hash marks and every angle for arcs. Marks with the same number (single, double, triple) indicate congruent groups. Always extract that information first, then start the problem.

Going further

Lesson 24 is the capstone of the Geometry strand. Real problems combine multiple relationships (vertical angles, supplementary pairs, corresponding angles across parallels, and corresponding parts of congruent polygons) in a single chain of reasoning. The correspondence rule you just learned shows up as one link in those chains: identify which angles match across two congruent polygons, then use the resulting equality to compute the next angle.

In Grade 8, you'll learn that for triangles you don't actually need to check all six corresponding parts. Theorems like SAS (two sides and the included angle) and ASA (two angles and the included side) prove congruence with just three pieces of information. Those theorems work because of the correspondence rule in this lesson; they're shortcuts that only make sense once you know how the matching is defined.