Basic Concepts of Triangle
- A closed figure formed by three intersecting lines is called a Triangle (‘Tri’ means ‘three’).
- It has three sides, three angles and three vertices.
- PQ, QR, PR are the three sides
- ∠P, ∠Q, ∠R are the three angles
- P,Q,R are three vertices.
- There are three types of triangles on the basis of the length of the sides.
- Scalene Triangle - All sides of the triangle are of different length.
- Isosceles Triangle- Two sides of the triangle are of equal length.
- Equilateral Triangle - All three sides of the triangle are of equal length.
- There are three types of triangles on the basis of angles.
- Acute angled Triangle - All three angles of the triangle have measure < 90°
- Obtuse angled Triangle - One angle of the triangle have measure > 90°
- Right angled Triangle - One angle of the triangle have measure = 90°
Meaning of Congruence
- If the shape and size of two figures are same then, those two figures are said to be Congruent.
- The symbol of congruent is “≅”.
Congruence of Triangles
A triangle will be congruent if its corresponding sides and angles are equal.
Example: In the figure shown below, ∆DEF is congruent to ∆ABC
What is CPCT ?
- CPCT stands for ‘corresponding parts of congruent triangles’.
- ‘Corresponding parts’ means corresponding sides and corresponding angles of the triangles.
- According to CPCT,
if two or more triangles are congruent to one another, then all of their corresponding parts are equal.
Criteria for Congruence of triangles
- SAS (Side-Angle-Side) Congruence Rule:
Two triangles are congruent if two sides and one included angle in a given triangle are equal to the corresponding two sides and one included angle in another triangle.
Example :
Given:
AB = PR,
AC = PQ and
∠ QPR = ∠ BAC,
then , by SAS Congruence Rule △ABC ≅△ PQR.
ASA (Angle-Side-Angle) Congruence Rule:
Two triangles are congruent if two angles and the included sides of one triangle are equal to corresponding two angles and the included side of the other triangle.
Given:
∠ BAC = ∠ PRQ,
AC=QR
∠ ACB = ∠ PQR.
then , by ASA Congruence Rule △ABC ≅△ PQR
AAS (Angle-Angle-Side) Congruence Rule:
Two triangles are congruent if their corresponding two angles and one non-included side are equal.
Given:
∠ BAC = ∠ QPR,
∠ ACB = ∠ RQP and
AB = QR,
then ,by AAS Congruence rule △ABC ≅△ PQR
SSS (Side-Side-Side) Congruence Rule:
Two triangles are congruent if three sides of one triangle are equal to the corresponding sides of the other triangle.
Given:
AB = PR,
AC = QP, and
BC = QR.
then, by SSS Congruence Rule △ABC ≅△ PQR.
RHS Congruence Rule:
If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangles, then the two triangles are congruent.RHS stands for Right angle – Hypotenuse – Side.
Given:
AC=PR (Hypotenuse)
∠ ABC = ∠ PQR (Right Angle)
BC = QR,
then, by RHS Congruence Rule △ABC ≅△ PQR.
Why SSA and AAA congruency rules are not valid?
- SSA or ASS test is not a valid test for congruence as the angle is not included between the pairs of equal sides.-
- The AAA test also is not a valid test as even though 2 triangles can have all three same angles, the sides can be of differing lengths. This becomes a test for similarity (AA).
Some Properties of a Triangle
- 1. Two angles opposite to the two equal sides of an isosceles triangle are also equal.
2. Two sides opposite to the equal angles of the isosceles triangle are also equal. This is the converse of the above theorem.
3. If two sides of a triangle are unequal then the longer side has the greater angle opposite it. Thus, we can say that angle opposite to the shorter side of a triangle is smaller.
For example, in the given triangle, AC > AB, therefore ∠ABC > ∠ACB.
4. If two angles of a triangle are unequal then the greater angle has the longer side opposite it.Thus, we can say that the smaller angle has the shorter side opposite it.
For example, in the given figure, ∠BAC > ∠ACB, therefore BC > AB.
5. The sum of any two sides of a triangle will always be greater than the third side.