congruence of triangles

Congruent Figures: When two geometric figures have the same size and shape they are said to be congruent. The symbol used to denote congruence is  \(\cong\)

Two congruent figures are identical or they are equal in all respects.

Two squares are congruent if they have the same ______. Solution: sides. A square is a figure with four equal straight sides and four right angles, hence the only property required to make two squares congruent is they have equal sides.

In triangles that are congruent, the six elements - three sides and three angles of the one are respectively equal to the six elements of the other. 

Conditions for congruency of triangles

 

Side-angle-side(SAS)​​​​​​​ Axiom  

If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by the Side-angle-side rule.

Note: In SAS,  criteria for the equality of the included angle is essential.

Angle-Side-Angle or Angle-Angle-Side Axiom (Two angles, corresponding sides)

Angle, Side, Angle(ASA) states that two triangles are congruent if they have an equal side contained between corresponding equal angles. Angle, Angle, Side(AAS) states that if the vertices of two triangles are in one-to-one correspondence such that two angles and the side opposite to one of them in one triangle are equal to the corresponding angles and the non-included side of the second triangle, then the triangles are congruent.

Note: The two equal sides must be corresponding sides.

Side-Side-Side Axiom (Three sides)

If all three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by Side-Side-Side (SSS) rule.

Right angle, hypotenuse, and sides (RHS) Axiom

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

Note:

When we want to say that a given triangle, like triangle ABC, is congruent to another triangle, like triangle \(DEF\), the order of the vertices in the name makes a big difference. When two triangles are written this way, ABC and \(DEF\), it means that vertex A corresponds with vertex D, vertex B with vertex E, and so on. These relationships aren't especially important when triangles aren't congruent or similar. But when they are congruent, the one-to-one correspondence of triangles determines which angles and sides are congruent.

 

1. If \(\bigtriangleup ABC \cong\, \bigtriangleup XYZ\), write the parts of \(\bigtriangleup XYZ\) that correspond to ∠B, BC, ∠C. Solution: ∠B = ∠Y, BC = YZ, ∠C = ∠Z (find the corresponding vertex for A, B and C in triangle XYZ) 2. If two triangles are congruent, what can you say about their area and perimeter? Solution:  Perimeter and area of both triangles are equal. As perimeter is equal to the sum of the three sides of a triangle, therefore as both the triangles are of equal sides their perimeter is also the same. The area of a triangle is equal to half of the base times height, i.e. A = 1/2 × b × h. As the base and height of both triangles are equal hence they have equal areas.

---

Theorem on Isosceles triangle

If two sides of a triangle are equal, then the angles opposite to those sides are equal. 

If \(AB\) = AC, then ∠C = ∠B

Conversely, if two angles of a triangle are equal, the sides opposite those angles are also equal. 

Example: Find the lettered angles in below figure - 

Solution:
In \(\bigtriangleup ADB\), ∠A = ∠D as AB = BD (If two sides of a triangle are equal, then the angles opposite to those sides are equal.) 
In \(\bigtriangleup DCB\), ∠C = ∠x as CD = BD (If two sides of a triangle are equal, then the angles opposite to those sides are equal.) 
∠ADB = 180 - 108 = 72° ⇒ ∠A = 72° 
Therefore ∠y = 180 − (72 + 72) ⇒ ∠y = 36°
∠x + ∠C + 108 = 180 
2∠x = 180 − 108 ⇒ ∠x = 36°