NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 6 (For CBSE, ICSE, IAS, NET, NRA 2022)
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Triangles
| S. no. | Term | Descriptions |
| 1 | Congruence | Two Geometric figure are said to be congruence if they are exactly same size and shape Symbol used is Two angles are congruent if they are equal Two circle are congruent if they have equal radii Two squares are congruent if the sides are equal |
| 2 | Triangle Congruence | Two triangles are congruent if three sides and three angles of one triangle is congruent to the corresponding sides and angles of the other Corresponding sides are equal Corresponding angles are equal We write this as The above six equalities are between the corresponding parts of the two congruent triangles. In short form this is called C. P. C. T We should keep the letters in correct order on both sides |
| 3 | Inequalities in Triangles | (1) In a triangle angle opposite to longer side is larger (2) In a triangle side opposite to larger angle is larger (3) The sum of any two sides of the triangle is greater than the third side In triangle ABC |
Different Criterion for Congruence of the Triangles
| N | Criterion | Description | Figures and expression |
| 1 | Side angle Side (SAS) congruence | Two triangles are congruent if the two sides and included angles of one triangle is equal to the two sides and included angle It is an axiom as it cannot be proved so it is an accepted truth ASS and SSA type two triangles may not be congruent always |  Then |
| 2 | Angle side angle (ASA) congruence | Two triangles are congruent if the two angles and included side of one triangle is equal to the corresponding angles and side It is a theorem and can be proved |  Then |
| 3 | Angle-angle side (AAS) congruence | Two triangles are congruent if the any two pair of angles and any side of one triangle is equal to the corresponding angles and side It is a theorem and can be proved |  Then |
| 4 | Side-Side-Side (SSS) congruence | Two triangles are congruent if the three sides of one triangle is equal to the three sides of the another |  Then |
| 5 | Right angle hypotenuse side (RHS) congruence | Two right triangles are congruent if the hypotenuse and a side of the one triangle are equal to corresponding hypotenuse and side of the another | If following condition Then |
Some Important Points on Triangles
| Terms | Description |
| Orthocenter | Point of intersection of the three altitude of the triangle |
| Equilateral | Triangle whose all sides are equal and all angles are equal to 600 |
| Median | A line Segment joining the corner of the triangle to the midpoint of the opposite side of the triangle |
| Altitude | A line Segment from the corner of the triangle and perpendicular to the opposite side of the triangle |
| Isosceles | A triangle whose two sides are equal |
| Centroid | Point of intersection of the three median of the triangle is called the centroid of the triangle |
| In center | All the angle bisector of the triangle passes through same point |
| Circumcenter | The perpendicular bisector of the sides of the triangles passes through same point |
| Scalene triangle | Triangle having no equal angles and no equal sides |
| Right Triangle | Right triangle has one angle equal to 900 |
| Obtuse Triangle | One angle is obtuse angle while other two are acute angles |
| Acute Triangle | All the angles are acute |
Similarity of Triangles
| S. no | Points |
| 1 | Two figures having the same shape but not necessarily the same size are called similar figures. |
| 2 | All the congruent figures are similar but the converse is not true. |
| 3 | If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points. then the other two sides are divided in the same ratio |
| 4 | If a line divides any two sides of a triangle in the same ratio. Then the line is parallel to the third side. |
Different Criterion for Similarity of the Triangles
| N | Criterion | Description | Expression |
| 1 | Angle-Angle angle (AAA) | Two triangles are similar if corresponding angle are equal similarity | If following condition Then Then |
| 2 | Angle-angle | Two triangles are similar if the two If following (AA) similarity corresponding angles are equal as condition by angle property third angle will be also equal | If following condition Then Then Then |
| 3 | Side-side-side (SSS) Similarity | Two triangles are similar if the sides of one triangle is proportional to the sides of other triangle | If following Condition Then Then |
| 4 | Side-Angle- Side (SAS) similarity | Two triangles are similar if the one angle of a triangle is equal to one angle of other triangles and sides including that angle is proportional | If following Condition And Then |
Area of Similar Triangles
If the two triangle ABC and DEF are similar
Then
Pythagoras Theorem
| S. no | Points |
| 1 | If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse. Then the triangles on both sides of the perpendicular are similar to the whole triangle and to each other. |
| 2 | In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem) . |
| 3 | If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle |
Arithmetic Progression
| S. no | Terms | Description |
| 1 | Arithmetic Progression | An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant Examples (1) 1, 5, 9,13, 17 … (2) 1, 2,3, 4,5 … |
| 2 | Common difference of the AP | the difference between any successive members is a constant and it is called the common difference of AP (1) If are the terms in AP then (2) We can represent the general form of AP in the form Where a is first term and d is the common difference |
| 3 | term of Arithmetic Progression | term |
| 4 | Sum of nth item in Arithmetic Progression | Or |
Trigonometry
| S. no | Terms | Description |
| 1 | What is Trigonometry | Trigonometry from Greek trigonon, ( “triangle,” and metron, “measure” ) is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees) . The applicability to non-right-angle triangles exists, but since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus, the majority of applications relate to right-angle triangles |
| 2 | Trigonometric Ratio՚s | In a right-angle triangle ABC where  We can define following term for angle A Base: Side adjacent to angle Perpendicular: Side Opposite of angle Hypotenuse: Side opposite to right angle We can define the trigonometric ratios for angle A as Notice that each ratio in the right-hand column is the inverse, or the reciprocal, of the ratio in the left-hand column. |
| 3 | Reciprocal of functions | The reciprocal of sin A is cosec A; and vice versa. The reciprocal of cos A is sec A And the reciprocal of tan A is cot A These are valid for acute angles. We can define tan And Cot A |
| 4 | Value of sin and cos | Is always less 1 |
| 5 | Trigonometric ration from another angle | We can define the trigonometric ratios for angle C as |
| 6 | Trigonometric ratios of complimentary angles | |
| 7 | Trigonometric identities |
Trigonometric Ratios of Common Angles
We can find the values of trigonometric ratio՚s various angle
| Angles (A) | Cos A | Tan A | Cosec A | Sec A | Cot A | |
| 0 | 1 | 0 | Not defined | 1 | Not defined | |
| 2 | ||||||
| 1 | 1 | |||||
| 2 | ||||||
| 1 | 0 | Not defined | 1 | Not defined | 0 |