Congruent Triangles

Congruent figures are equal in all respects.

So sides, angles and area are all equal. Thus, congruent figures can be made to coincide or fit exactly over each other so that all sides and vertices correspond.

Many problems in Geometry can be solved if we can prove that two or more figures are congruent to each other. Once figures have been shown to be congruent, we can describe equality relations between them and so solve for unknown quantities.

Theorems on Triangle Congruence:

There are really only three theorems about triangle congruence although most geometry books mention four. The additional theorem is just a special case of one of the three originals.

Though in recent times, students have been taught to remember the theorems through the use of letters such as SAS or SSS, I highly recommend that you learn the words associated with the letters in order to really understand the theorem. Besides, words make more sense to us than letters do.

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Side, Angle, Side (SAS)

If two sides and the contained angle of one triangle
are respectively equal to two sides and the contained angle of another,
the triangles are congruent or equal in all respects.

Given: ABC and DEF are two triangles in which AB = DE, AC = DF and Angle A = Angle D.

Req'd to prove: ( means "is congruent to")

Proof:

Let's assume that we can pick up Triangle ABC and place it on Triangle DEF so that
vertex A falls on vertex D and AB falls along DE. Since AB = DE, (given),
vertex B must fall on vertex E.

Since Angle A = Angle D, (given), AC will fall along DE and
since AC = DE, (given), vertex C must fall on vertex F.

Therefore BC coincides with EF and Triangle ABC coincides with Triangle DEF.

Therefore .

Since congruence means equal in all respects, we can say that Angle B = Angle E,

Angle C = Angle F, and BC = EF.
Also, the area of Triangle ABC = the area of Triangle DEF.

Note: Make sure the equal angles are contained by the equal sides.

Example:

The triangles on the left are congruent by SAS because the equal angles
are contained by the equal sides in the two triangles.

The triangles on the right are not congruent by SAS because the equal angles
are not contained by the equal sides in the two triangles.

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Side, Side, Side (SSS)

If the three sides of one triangle are respectively equal
to the three sides of another, the triangles are congruent.

Given: ABC and DEF are triangles in which AB = DE, AC = DF and BC = EF.
Req'd to Prove: .

Proof: Let's assume that we can pick up Triangle ABC and place it on Triangle DEF
so that vertex B falls on vertex E and BC falls along EF.
Since BC = EF, (given), C must fall on F.

Let A fall at G on the side of EF remote from D. Join DG.
Since DE = EG (given), Angle EDG = Angle EGD (isosceles triangle theorem)
Similarly, since DF = FG (given), Angle FDG = Angle FGD (isosceles triangle theorem).
So Angle EDG + Angle FDG = Angle EGD + Angle FGD (the sums of equals are equal)
Therefore Angle EDF = Angle EGF.
Therefore (Side, Angle, Side)
But Triangle EGF is really a reproduction of Triangle ABC,
therefore .

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Angle, Side, Angle (ASA)

If two angles and a side of one triangle are respectively equal to
two angles and the corresponding side of another,
the triangles are congruent.

Given: ABC and DEF are triangles in which Angle A = Angle D,
Angle B = Angle E and BC = EF.
Req'd to Prove: .
Proof:

Since Angle A = Angle D and Angle B = Angle E, (given),
Angle C = Angle F ( angle sum in a triangle = 180°).
Now, we apply Triangle ABC to Triangle DEF so that B falls on E
and BC falls along EF.
Since BC = EF (given), C falls on F.
Since Angle B = Angle E, (given), BA falls along ED with A falling on ED or ED produced.
And since Angle C = Angle F, (proof), CA falls along FD with A on FD or FD produced.
Thus, A must fall on D which is the only point common to ED and FD.
So, Triangle ABC coincides with Triangle DEF.
Therefore, Triangle .

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We now have three theorems with which to prove triangles are congruent.
There is one more such theorem that's a special case of the Side, Side Side theorem (SSS). It's known as the Right Triangle Congruence Theorem and its proof is based on the Pythagorean Theorem on right triangles.

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Notation: In a triangle with vertices A, B, and C, the lengths of the sides are denoted by a, b, and c with a the length of the side opposite vertex A etc. as shown in the diagram.
Also, the hypotenuse of a right-angled triangle is the side opposite the right angle.

The Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle,
the square on the hypotenuse is equal to the sum of the squares
of the lengths of the other two sides.

So, in the right-angled triangle ABC, c² = a² + b²

This statement can always be written in its two other forms:

c² – a² = b²

c² – b² = a²

So, given the lengths of two sides of a right triangle, we can always find the length of the third side. Note that the hypotenuse is always the longest side.

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Hypotenuse and One Side (HS)

If the hypotenuse and one side of a right-angled triangle
are respectively equal to the hypotenuse and one side of another
right-angled triangle, the triangles are congruent.

So this is just a special case of the side, side, side theorem since
the Pythagorean Theorem easily shows that BC = EF.

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Now we're ready to tackle proving deductions. Do the Practice questions in this lesson, then study the lesson titled Proving Deductions in which there's additional questions on congruent triangles in the Practice section.

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Practice

1) State whether the triangles are congruent (isometric) and state why or why not.

( solution )

2)

(solution a ) (solution b)

3) What additional information is needed for these triangles to be congruent?
(2 answers in (a))

(solution)

4) Pete and Repete are organizing a sailboat race. They're out on the water with the mother of all floating protractors, some rope and 4 buoys. They must create the race course for 2 boats like in the diagram. The race goes from A to B with each sailboat going around the buoy on its side of the course.

They set a buoy at point A, the starting point, another at point B, from which they measure two equal angles on either side of the line AB. Then they place the remaining 2 buoys at points C and D which are both 350 meters from B.

a) Which congruence theorem assures them that the two courses are equal in length?

b) Write a full and justified proof to show that AC = AD.

(solution)

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Solutions

1)

a) congruent by SAS

b) congruent by ASA

2) a)

If we can show that Angle ACB ( x + z ) = Angle EBC ( y + z ),
the triangles would be congruent by Side Angle Side (SAS).

Proof: triangles ADC, EDB and DCB are all isosceles (equal sides given)

This means Angle DAC = Angle DCA ( x) ; Angle DBE = Angle DEB, ( y )
and Angle DCB = Angle DBC ( z ); (isosceles triangle thm.)

Now, if we could show that x = y, we could show that ( x + z ) = ( y + z )

by Side, Side Side (SSS)

So x = y and therefore x + z = y + z

This makes by Side Angle Side (SAS)

b) In triangles ACE and ADE:

CE = ED (given)

Angle AEC = Angle AEC = 90° (given)

AE = AE (common side or reflexive property)

(SAS)

3)

a) Angle A = Angle D for SAS BC = FE for SSS

b) GH = KL for ASA

4) a) Side, Angle, Side (SAS)

b)

In triangles ABC and ABD:

BC = BD = 350 (given)

Angle ABC = Angle ABD (given)

AB = AB (common side or reflexive property)

(SAS).

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