Congruent Triangles |

**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.

.

**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.

.

**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:** .

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 .

.

**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 .

.

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.

.

**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**, ** c ^{2} = a^{2} + b^{2}**

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

**c ^{2} – a^{2} = b^{2}**

**c ^{2} – b^{2}** =

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.

.

**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**.

.

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.

.

**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*)

.

**Solutions**

a) congruent by SAS | b) congruent by ASA |

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 (

and Angle DCB = Angle DBC (

- Now, if we could show that

- by Side, Side Side (SSS)

- So

- This makes

b) In triangles ACE and ADE:

- CE = ED (given)

- Angle AEC = Angle AEC = 90° (given)

- AE = AE (common side or reflexive property)

- (SAS)

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).

.

( Plane Geometry MathRoom Index )

(*all content **© MathRoom Learning Service; 2004 - *).