Triangle Congruence Proofs — Print

Six parts make a triangle, but you never need all six. Three of the right kind are enough.

What the practice questions cover

These questions use 15 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Corresponding sides — Sides in matching positions in two similar figures.
Vertex — A flat surface; a line where two faces meet; a corner point.
Vertical angles — The opposite pair formed by two crossing lines — always equal.
Base — The number being multiplied. In 53 the base is 5.
Hypotenuse — The side opposite the right angle — always the longest.
Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
Corresponding parts — The pieces that match up: A with D, side AB with side DE, and so on.
Parallel — Same slope, different intercept.
Prove — The statement you must arrive at. Always the last line.
Included angle — The angle between two named sides. The A in SAS must be included.
Included side — The side between two named angles. The S in ASA must be included.
CPCTC — Corresponding Parts of Congruent Triangles are Congruent — used after proving congruence.
Reflexive property — Anything is congruent to itself. A shared side is congruent to itself.
Ambiguous case — Two sides and a non-included angle, which can describe two different triangles.
Combination — A selection where order does not matter. C(n,r) = n!r!(n-r)!.

How much do you have to tell someone before their triangle must match yours exactly?

One side? Hopeless — they can build endless triangles on it.

All three sides? Now it is locked. Try it with three straws: there is only one triangle you can make, however you twist them.

So three sides are enough. That shortcut is called SSS.

But three angles are not enough — two triangles can share all three angles and be wildly different sizes.

Four combinations do lock a triangle: SSS, SAS, ASA and AAS. One more, HL, works only for right triangles. Everything else leaves room to wriggle.

The four that work

SSS: three pairs of sides. SAS: two sides and the angle between them. ASA: two angles and the side between them. AAS: two angles and a side not between them.

AAS works because knowing two angles gives you the third for free — so AAS is really ASA in disguise.

For right triangles there is a fifth: HL, a matching hypotenuse and one leg.

What "included" means

In SAS the angle must sit between the two sides. Two sides plus an angle somewhere else is SSA, which does not work — and that single word is the whole difference.

The two that do not

AAA gives similar triangles, not congruent ones. All the angles match and the sizes need not.

SSA is the ambiguous case: swing the unattached side and it can meet the far line in two different places, giving two different triangles.

If your proof needs one of these, you have not got enough — go back and find another given.

Finding the third part

A proof usually hands you two parts and hides the third. The commonest hiding places are worth knowing.

A shared side is congruent to itself — cite the reflexive property. A pair of vertical angles is automatically congruent.

Parallel lines in the figure give you alternate interior or corresponding angles. A midpoint gives two equal segments.

Mark the diagram

Tick the givens onto the picture, then look for the free third part. It is almost always a shared side or a pair of vertical angles, and once marked the proof is visible.

CPCTC: what congruence buys you

Once two triangles are proved congruent, every pair of corresponding parts is congruent. That is CPCTC.

This is usually the point of the proof. You want to show two segments are equal, so you prove the triangles containing them are congruent, then cite CPCTC.

The order matters: CPCTC comes after congruence, never before. Using it to prove congruence is circular.

Isosceles triangles

If two sides of a triangle are congruent, the angles opposite them are congruent. This is the base angles theorem.

Its converse is also true and also useful: equal base angles force equal sides.

Both are proved by drawing the bisector from the apex and using SAS — a good example of adding a line to create triangles that were not there.

Worked examples

Choosing the right shortcut

In △ ABC and △ DEF: AB ≅ DE, ∠ B ≅ ∠ E, BC ≅ EF. Which shortcut applies?

List what you have in order around the triangle. Side AB, angle B, side BC.
The angle sits between the two sides. ∠ B is formed by AB and BC.
So it is SAS, not SSA. Included angle — the shortcut applies.
△ ABC ≅ △ DEF by SAS. Letters in matching order.

SAS

Finding the hidden third part

Two triangles share side BD. You are given AB ≅ CB and ∠ ABD ≅ ∠ CBD. Prove △ ABD ≅ △ CBD.

AB ≅ CB — Given One side.
∠ ABD ≅ ∠ CBD — Given The angle between them.
BD ≅ BD — reflexive property The shared side is the free third part.
△ ABD ≅ △ CBD by SAS Side, included angle, side.

Congruent by SAS

Using CPCTC

Having proved △ ABD ≅ △ CBD, prove AD ≅ CD.

△ ABD ≅ △ CBD — proved above Congruence first.
AD ≅ CD — CPCTC AD and CD are corresponding sides.
Check the correspondence: A→C, B→B, D→D. So AD really does pair with CD.

AD ≅ CD by CPCTC

Words to know

Included angle: The angle between two named sides. The A in SAS must be included.
Included side: The side between two named angles. The S in ASA must be included.
CPCTC: Corresponding Parts of Congruent Triangles are Congruent — used after proving congruence.
Reflexive property: Anything is congruent to itself. A shared side is congruent to itself.
Hypotenuse-Leg: HL: for right triangles only, a matching hypotenuse and one leg are enough.

Mistakes to watch for

Using SSA as though it were a shortcut. Two sides and a non-included angle can give two different triangles — the ambiguous case — so it proves nothing. Check that the angle in SAS really sits between the two sides.

Treating AAA as congruence. Equal angles give similar triangles, which may be any size: a small triangle and a huge one can have identical angles. You need at least one side to fix the scale.

Citing CPCTC before proving congruence. CPCTC is what congruence gives you, so using it to establish congruence is circular. It can only ever appear after the congruence line.

Writing the letters in the wrong order. △ ABC ≅ △ DFE claims B pairs with F, which is a different and usually false statement. The order is the correspondence.

SSA is deliberately missing from the list, and the reason is worth seeing: two sides and a non-included angle can produce two different triangles.

It is called the ambiguous case, and it is why the law of sines sometimes gives two answers later on.

Some textbooks avoid writing SSA at all, because reversed it spells something the authors would rather not print.

Problem 1

Given three pairs of congruent sides, which shortcut applies?

Worked answer

SSS

Three sides, no angles.

That is SSS.

Problem 2

Two angles and the side between them are congruent. Which shortcut?

Worked answer

ASA

The side sits between the two angles.

That is ASA.

Problem 3

Which combination does NOT prove congruence?

Worked answer

SSA

SSA can give two different triangles.

It proves nothing.

Problem 4

Two triangles share side BD. What reason gives BD ≅ BD?

Worked answer

reflexive property

BD ≅ BD by the reflexive property.

Problem 5

After proving △ ABC ≅ △ DEF, what reason lets you write ∠ B ≅ ∠ E?

Worked answer

CPCTC

B pairs with E in the correspondence.

So ∠ B ≅ ∠ E by CPCTC.

Problem 6

AB ≅ DE, ∠ B ≅ ∠ E, BC ≅ EF. Name the shortcut.

Worked answer

SAS

∠ B is formed by AB and BC.

Included angle, so SAS.

Problem 7

∠ A ≅ ∠ D, ∠ B ≅ ∠ E, BC ≅ EF. Name the shortcut.

Worked answer

AAS

BC is not between ∠ A and ∠ B.

Two angles and a non-included side: AAS.

Problem 8

In △ ABC, AB ≅ AC and m∠ B = 52°. Find m∠ C.

Worked answer

Equal sides give equal opposite angles.

m∠ C = 52°

Problem 9

In △ ABC, AB ≅ AC and m∠ A = 40°. Find m∠ B.

Worked answer

180 - 40 = 140

140 ÷ 2 = 70°

Problem 10

HL may be used on

right triangles only
any triangle
isosceles triangles only
obtuse triangles only

Worked answer

right triangles only

Hypotenuse — so there must be a right angle.

HL is right triangles only.

Problem 11

Two triangles meet at a point so that a pair of angles are vertical. Are those angles congruent? Answer yes or no.

Worked answer

yes

No parallel lines needed.

Vertical angles are congruent by theorem.

Problem 12

Two triangles have all three angles congruent. Are they congruent? Answer yes or no.

Worked answer

AAA gives similar triangles.

They could be any size, so not necessarily congruent.

Problem 13

△ ABC ≅ △ DEF with AB = 3x + 1 and DE = 2x + 7. Find AB.

Worked answer

3x + 1 = 2x + 7, so x = 6

AB = 3(6) + 1 = 19

Problem 14

An isosceles triangle has a base angle of (3x)° and vertex angle (2x + 20)°. Find x.

Worked answer

3x + 3x + 2x + 20 = 180

8x = 160

x = 20

Problem 15

Given AB ≅ CB and ∠ ABD ≅ ∠ CBD, with BD shared. Name the shortcut proving △ ABD ≅ △ CBD.

Worked answer

SAS

Side AB, included angle ∠ ABD, side BD (reflexive).

That is SAS.

Problem 16

From △ ABD ≅ △ CBD, which side corresponds to AD?

Worked answer

AD pairs with CD.

AD ≅ CD by CPCTC.

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