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Congruent triangle tests

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Two mirror image roof brackets. Triangle ABC has side AB 7 cm, side BC 6.4 cm and angle B 50 degrees. Triangle DEF has side DE 7 cm, side EF 6.4 cm and angle E 50 degrees.ABCDEF
AB = DE
\(\displaystyle 7\ \mathrm{cm}\)
BC = EF
\(\displaystyle 6.4\ \mathrm{cm}\)
Angle B = angle E
\(\displaystyle 50^{\circ}\)

Two roof brackets have the measurements shown. Which statement follows directly from those measurements?

Year 9, Chapter 7, 4 difficulty levels, No calculatorBuild a worksheet

Congruent triangle tests

Congruent triangles have the same size and shape. Match corresponding vertices first, then check whether the information satisfies SSS, SAS, AAS or RHS

\[\triangle ABC\cong\triangle DEF\]
  1. Same size and shape Corresponding sides and angles are equal. Turning or reflecting a triangle does not change its size or shape
  2. Corresponding order In ABC congruent to DEF, A matches D, B matches E and C matches F. Follow the order, not the position on the page
  3. SSS Three pairs of corresponding sides are equal
  4. SAS Two corresponding side pairs and the angle between those sides are equal
  5. AAS or ASA Two corresponding angle pairs and any corresponding side are equal. The third angle is fixed by the 180° total, so ASA is equivalent
  6. RHS Both triangles are right-angled, with equal hypotenuses and one other corresponding side equal
  7. Insufficient data AAA fixes shape but not size. SSA does not generally fix one triangle. Equal area alone is also insufficient

Worked example

Triangle ABC has AB = 6 cm, BC = 8 cm and CA = 9 cm. Triangle DEF has DE = 6 cm, EF = 8 cm and FD = 9 cm. Which test and matching statement follow?

\[\triangle ABC\cong\triangle DEF\]
  1. \(\displaystyle AB=DE=6\ \mathrm{cm}\)The other pairs also match: BC = EF = 8 cm and CA = FD = 9 cm
  2. \(\displaystyle \mathrm{SSS}\)Three equal corresponding side pairs satisfy the SSS congruence test
  3. \(\displaystyle \triangle ABC\cong\triangle DEF\)The vertex order matches the side correspondence A to D, B to E and C to F.

Common mistakes

Mismatching corresponding vertices
✕\(\displaystyle \triangle PQR\cong\triangle LNM\)✓\(\displaystyle \triangle PQR\cong\triangle LMN\)
When PQ matches LM, QR matches MN and RP matches NL, P matches L, Q matches M and R matches N. Write those vertices in corresponding order
Using AAA as a congruence test
✕\(\displaystyle \mathrm{AAA}\)✓\(\displaystyle \mathrm{SSS}\)
AAA fixes the angles but not the size, while SSS fixes all three side lengths
Missing the included angle in SAS
✕\(\displaystyle \mathrm{SSA}\)✓\(\displaystyle \mathrm{SAS}\)
In SAS the known angle must lie between the two known corresponding sides

Difficulty levels

introMatch the order of corresponding vertices in a congruence statement
standardChoose SSS, SAS, AAS or RHS from the marked equal parts
challengeDecide whether a proposed test is sufficient and reject an AAA or SSA case
examSelect the test and the correctly ordered congruence statement from several conditions

Questions about this topic

What does SSS mean?+

SSS means side-side-side. Three pairs of equal corresponding side lengths are enough to prove triangle congruence

Why is AAA not a congruence test?+

AAA determines the angles and therefore the shape, but a triangle with the same angles can have a different size

What does the order in a congruence statement show?+

It shows the correspondence between vertices. The positions in the two triangle names must match

Is ASA also a valid test?+

Yes. Two angles determine the third, so an equal corresponding side between those angles works just as it does for AAS

Why does RHS require a hypotenuse?+

The hypotenuse is opposite the right angle. RHS requires that longest side and one other side to match, with both right angles established

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 7: Geometry and similarity

In this chapter

Angle facts and parallel linesTriangle angle sum, special triangles and the exterior angle theoremQuadrilateral angle sums and special-quadrilateral propertiesPolygon angle sums and regular polygonsCongruent triangle testsProve congruence and use it to prove a further resultEnlargement transformation and scale factor7G Enlargement and scale: Similar figures and scale factorsTests for similar triangles7H Similar triangle tests: Similar figures and scale factorsApplied similarity and scaleSimilar solids and scaling volumeDesign, test and refine geometrical algorithms