FG5–FG6, HG5–HG6 - Congruence and geometric proof

FG5–FG6, HG5–HG6 - Congruence and geometric proof

When does matching a few sides or angles guarantee that two whole triangles match? FG5 and HG5 give four congruence tests. Use them with angle and quadrilateral facts in FG6 and HG6 to justify results and distinguish a proof from a convincing-looking sketch.

Congruent triangles

Could two triangles have all the same angles but different sizes? Yes: an equilateral triangle of side 3 cm3\,\mathrm{cm} and one of side 5 cm5\,\mathrm{cm} both have three 60∘60^\circ angles. Matching angles alone does not fix size.

Congruent shapes

Congruent shapes have exactly the same size and shape. A rotation, reflection or translation can make one fit exactly over the other. Corresponding sides and corresponding angles are equal.

You need not compare every measurement. Each of these conditions fixes a triangle's size and shape, allowing a reflected version too.

TestRequired matching informationCondition that matters
SSSThree sidesMatch all three lengths
SASTwo sides and the angle between themThe angle must be included between those sides
ASATwo angles and a corresponding sideMatch the side's position; two angles fix the third
RHSRight angle, hypotenuse and one other sideBoth triangles must be right-angled

The hypotenuse is the side opposite the right angle. RHS uses this identified side, not whichever side happens to be drawn at the top. You do not need to calculate the remaining side to use RHS.

Why do these tests work? With SSS, two fixed distances from the endpoints of a fixed base locate the third vertex, up to reflection. With SAS, the two sides form a fixed opening. With ASA, fixed directions and one fixed length locate the remaining vertex. With RHS, a fixed hypotenuse and one leg fix a right triangle. By contrast, AAA fixes only shape, and two sides with a non-included angle (SSA) can give different triangles. SSA is not a general congruence test.

Worked example 1: SSS and SAS

Question. Triangle ABCABC has side lengths AB=5AB=5, BC=7BC=7, AC=8AC=8. Triangle PQRPQR has PQ=7PQ=7, QR=8QR=8, PR=5PR=5. Are they congruent? Separately, two triangles each have sides 4 cm4\,\mathrm{cm}, 6 cm6\,\mathrm{cm} and an included angle of 52∘52^\circ; are these congruent?

Plan. Match the stated measurements and choose the condition they establish.

For the first pair,

AB=RP,BC=PQ,AC=RQ.AB=RP,\qquad BC=PQ,\qquad AC=RQ.

All three sides match, so triangle ABCABC is congruent to triangle RPQRPQ by SSS. The order matches A↔RA\leftrightarrow R, B↔PB\leftrightarrow P, C↔QC\leftrightarrow Q.

For the second pair, two sides and the angle between those sides match, so they are congruent by SAS.

Answer. Both pairs are congruent, by SSS and SAS respectively.

Check. Each corresponding side agrees. In the SAS pair, the 52∘52^\circ angle is explicitly between the 44 and 66 sides; moving it elsewhere would remove the SAS justification.

Keep the goal of a definite triangle, but change the kind of information supplied.

Worked example 2: ASA and RHS

Question. Triangle DEFDEF has angles D=41∘D=41^\circ, E=68∘E=68^\circ and DE=9 cmDE=9\,\mathrm{cm}. Triangle XYZXYZ has X=41∘X=41^\circ, Y=68∘Y=68^\circ and XY=9 cmXY=9\,\mathrm{cm}. Are they congruent? Separately, two right triangles each have hypotenuse 13 cm13\,\mathrm{cm} and a shorter side 5 cm5\,\mathrm{cm}; are they congruent?

Plan. Check angle and side correspondence for ASA; identify the hypotenuse for RHS.

For the first pair,

∠D=∠X,DE=XY,∠E=∠Y.\angle D=\angle X,\qquad DE=XY,\qquad\angle E=\angle Y.

Thus DEFDEF and XYZXYZ are congruent by ASA. The equal sides join the matched angles.

For the second pair, the right angles, hypotenuses and one corresponding leg match, giving RHS.

Answer. Both pairs are congruent, by ASA and RHS respectively.

Check. Each third angle in the ASA pair is 180∘−41∘−68∘=71∘180^\circ-41^\circ-68^\circ=71^\circ. In the RHS pair, the 13 cm13\,\mathrm{cm} side is opposite the right angle in both triangles.

A proof must establish the whole condition, not just name it.

Watch out

“Two sides and an angle match, therefore SAS” misses the word included. Check that the named angle lies between the two known sides. Three equal angles also do not prove congruence: the opening equilateral triangles have different sizes.

Use a right angle to identify the correct test.

Angle proofs

A geometric proof is a chain of justified statements: given facts lead to a congruence condition, then congruence gives a new fact about matching sides or angles.

Use three letters to identify an angle precisely. In ∠BAC\angle BAC, the middle letter AA is the vertex. If a diagonal splits an angle, its two pieces need separate names. A shared side can be used in both triangles: write, for example, AC=CAAC=CA, common side.

[DIAGRAM: asset_name: Proofs using diagonals; asset_slug: quadrilateral-proof-diagonals; caption: A diagonal creates two triangles whose matching information can be justified.; description: The left panel shows a non-rectangular parallelogram ABCD with diagonal AC and opposite sides marked parallel. The right panel shows a non-square rhombus ABCD with diagonal AC and all four sides marked equal.]
A diagonal creates two triangles whose matching information can be justified.

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