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# Congruent Triangles

Triangles that have exactly the same size and shape are called congruent triangles.

### Corresponding parts

The parts of two triangles that have the same measure are referred to as corresponding parts. Corresponding parts of congruent triangles are equal.

### properties of congruent triangles

If two triangles are congruent, then their corresponding parts are equal and their corresponding angles are equal.

### Prove triangles congruent - ( SSS ) postulate

If each side of a triangle is congruent to corresponding side of another triangle, then the two triangles are congruent.  In the figure above, in triangle ABC and triangle RST, if AB = RS, BC = ST and AC = RT, then triangle ARC is congruent to triangle RST.

Note: 1. vertex A corresponding to vertex R, vertex B corresponding to vertex B and vertex C corresponding to vertex T. 2. AB corresponding to RS, BC corresponding to ST and CA corresponding to TR.

### Prove triangles congruent (SAS) postulate

If two sides and the angle between them in a triangle are congruent to corresponding parts in another triangle, then the two triangles are congruent.  In the figure above, in triangle ABC and triangle RST, if AB = RS, angle A = angle R and AC = RT, then triangle ABC is congruent to triangle RST.

Note: 1. two side of the angle A is AB and AC. Two sides of the angle R is RS and RT. 2. AB in triangle ABC corresponding to RS in triangle RST. Angle A in triangle ABC corresponding to angle R in triangle RST. AC in triangle ABC corresponding to RT in triangle RST. 3. when these three pairs of corresponding parts in two triangles are congruent, then these two triangles are congruent.

### Prove triangles congruent (ASA) postulate

If two angles and the side between them in a triangle are congruent to corresponding parts in another triangle, then the two triangles are congruent.  In th figure above, in triangle ABC and triangle RST, angle A = angle R, AB = RS and angle B = angle S, then triangle ABC is congruent to triangle RST.

Note: 1. angle A in triangle ABC corresponding to angle R in triangle RST. AB in triangle ABC corresponding to RS in triangle RST. angle B in triangle ABC corresponding to angle S in triangle RST. 2. AB is the side between angle A and angle B in triangle ABC. RS is the side between angle R and angle S in triangle RST.

### Prove triangles congruent (AAS) theorem

If two angles and a side not between them in a triangle are congruent to corresponding parts in another triangle, then the two triangles are congruent.  In the figure above, in triangle ABC and triangle RST, if angle A = angle R, angle B = angle S and BC = ST, then triangle ABC is congruent to triangle RST.

Note: 1. angle A in triangle ABC corresponding to angle R in triangle RST. Angle B in triangle ABC corresponding to angle S in triangle RST. BC in triangle ABC corresponding to ST in triangle RST. 2. BC is the side not between angle A and angle B. ST is the side not between angle E and angle S.

### Prove triangles congruent (HL) theorem

If the hypotenuse and a leg in a right triangle are congruent to corresponding parts in another right triangle, then the two triangles are congruent.  In the figure above, in right triangle ABC and right triangle RST, if BC = ST and BA = SR, then right triangle ABC is congruent right triangle RST.

Note: 1. BC is the hypotenuse of the right triangle ABC. ST is the hypotenuse of the right triangle RST. 2. BA is a leg in right triangle ABC. ST is a leg in right triangle RST. 3. BA in right triangle ABC corresponding to SR in right triangle RST. BC in right triangle ABC corresponding to ST in right triangle RST.

Example 1

Given: in triangle ABC, AB = AC. AD is the bisector of the angle BAC. Prove: BD = CD Proof
since AD is the angle bisector of the angle BAC (given),
so angle 1 = angle 2 (Property of an angle bisector)
so BD = CD (in two congruent triangles, congruent angles opposite congruent sides.)

Note: 1. AB and AC are a pair of corresponding sides. angle 1 and angle 2 are a pair of corresponding angles. AD is the common side in both triangle BAD and triangle CAD. 2. Angle 1 opposite BD and angle 2 opposite CD. When triangle BAD and triangle CAD are congruent and angle 1 = angle 2, there is BD = CD (congruent angles opposite congruent sides) 3. In congruent triangles, corresponding angles are equal and corresponding sides are equal.

Example 2

Given: AB is perpendicular to BE and AC is perpendicular to CE. If AB = AC, prove BE = CE. Solution
Since AB is perpendiculat to BE (given), so triangle ABE is a right triangle and angle B = 90o
Since AC is perpendicular to CE (given), so triangle ACE is a right triangle and angle C = 90o
In right triangle ABE and right triangle ACE,
since AB = AC (given), AE in right triangle ABE = AE in right triangle ACE (common side)
so right triangle ABE is congruent to right triangle ACE (LH theorem)
so BE = CE (In two congruent triangles, corresponding sides are congruent.)

Note: 1. AB and AC are a pair of corresponding sides. AE in right triangle ABE and AE in right triangle ACE are a pair of corresponding sides. Angle B and angle C are a pair of corresponding angles. 2. In congruent triangles ABE and ACE, BE and CE are a pair of corresponding sides. 3. corresponding parts of two congruent triangles are congruent.

Example 3

In the figure below, if angle ABC = angle ACB and AD = AE. Prove BE = CD. Proof
Since angle ABC = angle ACB (given), so AB = AC (In a triangle, congruent angles opposite congruent sides.)
In triangle ABE and triangle ACD,
since AB in riangle ABE = AC in triangle ACD,
angle A in triangle ABE = angle A in triangle ACD (common angle),
AE in triangle ABE = AD in triangle ACD (given),
so triangle ABE is congruent to triangle ACD (SAS postulate)
so BE in triangle ABE = CD in triangle ACD (In two congruent triangles, congruent angles opposite congruent sides.)

Note: 1. in triangle ABC, angle ABC opposite AC and angle ACB opposite AB. Since angle ABC = angle ACB, so AC = AB. 2. AB in triangle ABE and AC in triangle ACD are a pair of corresponding sides. Angle A in triangle ABE and angle A in triangle ACD are a pair of corresponding angles. AE in triangle ABE and AD in triangle ACD are a pair of corresponding sides. 3. corresponding parts of two congruent triangles are congruent.