Geometric Twins · Lesson 3 of 10
Conventions to Express Congruence
“Match vertices in order so every side and angle comparison means the right thing.”
• Identify corresponding vertices, sides, and angles in congruent triangles. • Read and write a congruence statement with the correct vertex order. • Generate equivalent statements without changing the correspondence. • Use a common side and SSS in a rectangle split by a diagonal.
Conventions to Express Congruence
Saying that two triangles fit is only the beginning. We need to say which corner fits which corner. These paired corners are corresponding vertices. When A overlaps X, B overlaps Y, and C overlaps Z, side AB overlaps XY and angle ABC overlaps XYZ. The matching is determined by the fit; it is not chosen just because letters look convenient.
The vertices, sides, and angles that overlap when congruent figures are superimposed.
| Vertex match | Sides incident at the vertex | Angle match |
|---|---|---|
| A ↔ X | AB ↔ XY; AC ↔ XZ | angle A ↔ angle X |
| B ↔ Y | BA ↔ YX; BC ↔ YZ | angle B ↔ angle Y |
| C ↔ Z | CA ↔ ZX; CB ↔ ZY | angle C ↔ angle Z |
The statement triangle ABC ≅ triangle XYZ records this pairing in order. The symbol ≅ means “is congruent to.” First matches first, second matches second, and third matches third. A side name can be reversed, such as AB or BA, but the whole triangle statement must still preserve all three vertex matches.
Problem
If triangle AIR ≅ triangle FLY, list the corresponding parts.
- 1.Vertices: A ↔ F, I ↔ L, and R ↔ Y, because the positions in the names match.
- 2.Sides: AI = FL, IR = LY, and AR = FY.
- 3.Angles: angle A = angle F, angle I = angle L, and angle R = angle Y.
Rename both triangles together
You may start the triangle name at a different vertex or reverse its direction. The corresponding name on the other side must change in the same way. For example, from ABC ≅ XYZ, ACB ≅ XZY is correct: A still matches X, C still matches Z, and B still matches Y. ACB ≅ XYZ changes the pairing and is not justified by the original statement.
Problem
Given triangle HEN ≅ triangle BIG, write the other five orders with the first triangle on the left.
- 1.Keep the matches H ↔ B, E ↔ I, and N ↔ G.
- 2.Starting at H in the other direction gives HNE ≅ BGI.
- 3.Starting at E gives EHN ≅ IBG and ENH ≅ IGB.
- 4.Starting at N gives NHE ≅ GBI and NEH ≅ GIB. These are five rewritten statements of the same matching.
The diagonal of a rectangle
Let ABCD be a rectangle named around its boundary, with A top-left, B top-right, C bottom-right, and D bottom-left. Draw diagonal BD. The two triangles share BD. Opposite rectangle sides also agree: AB = CD and AD = CB. These three equalities give SSS, but we must still write the vertex order correctly.
Problem
Prove triangle ABD ≅ triangle CDB.
- 1.AB = CD because opposite sides of a rectangle are equal.
- 2.AD = CB for the same reason, and BD = DB because both names refer to the shared diagonal.
- 3.The match is A ↔ C, B ↔ D, and D ↔ B. Therefore ABD ≅ CDB by SSS.
If instead we paired B with B and D with D, AB would be paired with CB. A general rectangle may have unequal length and breadth, so that pairing need not work. A congruence statement must match the actual sides, not merely the shared endpoint letters.
Common vertices need not correspond to themselves. The two triangles in the rectangle share B and D, but their correct SSS pairing swaps those vertices.
Quiz
If ABC ≅ XYZ, which vertex corresponds to C?
If ABC ≅ XYZ, which side corresponds to AC?
Which statement preserves the pairing ABC ≅ XYZ?
What is the common side of triangles ABD and CDB?
In the rectangle above, which order matches triangle ABD?
Practice Problems
- Given triangle AIR ≅ triangle FLY, list three vertex, three side, and three angle matches.
- Write all six orders equivalent to ABC ≅ XYZ while keeping the first triangle on the left.
- Explain why ACB ≅ XYZ is not justified by ABC ≅ XYZ.
- Draw the rectangle ABCD with diagonal BD and write each SSS equality for ABD ≅ CDB.
- If PQR ≅ LMN and angle Q = 52°, what is angle M? State the reason.
- Triangles RED and JAM have RE = JA = 3.5 cm, ED = AM = 5 cm, RD = JM = 6 cm. Write the correct congruence statement.
Key Takeaways
• A congruence statement records a vertex pairing in its order. • Corresponding sides and angles are equal after congruence is established. • Rename both triangles together to preserve the pairing. • A common side supplies a useful equality for SSS. • The same named vertex can belong to both triangles without corresponding to itself.