Two circles are always congruent.

1. Triangle and its Types:

(i) A plane figure bounded by three lines in a plane is called a triangle.

(ii) A triangle, no two of whose sides are equal is called a scalene triangle.

(iii) A triangle whose two sides are equal is called an isosceles triangle.

(iv) A triangle whose all sides are equal is called an equilateral triangle.

(v) A triangle with one angle a right angle is called a right triangle.

2. Properties of Triangle:

(i) The sum of the three angles of a triangle is ${180}^{°}$.

(ii) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

(iii) Two figures are congruent if they are of the same shape and of the same size.

(iv) Angles opposite to equal sides of a triangle are equal.

(v) Sides opposite to equal angles of a triangle are equal.

(vi) Each angle of an equilateral triangle is ${60}^{°}$.

(vii) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles.

(viii) In an isosceles triangle altitude from the vertex bisects the base.

(ix) If the bisector of the vertical angle of a triangle bisects the base of the triangle, then the triangle is isosceles.

(x) If the altitudes of the triangle are equal, then it is equilateral.

(xi) In a triangle, the angle opposite to the longer side is larger.

(xii) In a triangle, the side opposite to the larger angle is longer.

(xiii) The sum of any two sides of a triangle is greater than the third side.

3. Congruent Figures:

(i) Two-line segments are congruent if their lengths are equal.

(ii) Two angles are congruent if they are of the same measure.

(iii) Two circles of the same radii are congruent.

(iv) Two squares of the same size are congruent.

4. Congruency of Triangles:

(i) If two triangles $ABC$ and $DEF$ are congruent under the correspondence $A\leftrightarrow D$, $B\leftrightarrow E$ and $C\leftrightarrow F,$ then we write $△ABC\cong △DEF$ or $△ABC\leftrightarrow △DEF$.

(ii) Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other triangle (SAS congruence criterion).

(iii) Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle. (ASA congruence criterion).

(iv) If any two angles and non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the triangles are congruent (AAS congruence criterion).

(v) If three sides of one triangle are equal to three of the other triangle, then the two triangles are congruent (SSS congruence criterion).

(vi) If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS congruence criterion).

5. Of all line segments that can be drawn to a given line, from a point, not lying on it, the perpendicular line segment is the shortest.