## All theorems | English medium. Grade 9,10,11

English medium, grade 10, grade 11, Grade 9 / September 4, 2021 July 13, 2022 / All theorems, All Theorems | English Medium. Grade 9, Grade 09, Grade 10, Grade 11

- Sum of the exterior angles of a polygon = 360°.

- The area of the square drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares drawn on the remaining two sides.

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

- The locus of points equidistant from two intersecting straight lines is the angle bisector of the angle formed by the intersection of the two lines.

- The sum of the three interior angles of a triangle is 180°.

- The locus of points that are at a constant distance from a straight line are the two straight lines parallel to it and at the given constant distance from it, on either side of it.

- The locus of points that are equidistant from two given points is the perpendicular bisector of the line joining the two points.

- The locus of points on a plane that is at a constant distance from a fixed point is a circle.

- A set of points satisfying one or more conditions is known as a locus.

- The method of solving a problem based on the value of a unit is called the unitary method.

- Two distinct quantities are said to be in direct proportion if they increase or decrease in the same ratio.

- When two straight lines are intersected by a transversal, if
- i) a pair of corresponding angles are equal or
- ii) a pair of alternate angles are equal or
- iii) the sum of a pair of allied angles is 180°, then the two straight lines are parallel to each other.

- When a transversal intersects a pair of parallel lines,
- i) the corresponding angles formed are equal,
- ii) the alternate angles formed are equal,
- iii) the sum of each pair of allied angles formed equals two right angles.

- A line intersecting two or more straight lines is known as a transversal.

- The sum of the magnitudes of the angles on a straight line is 180°.

- The sum of the adjacent angles formed by a straight line meeting another straight line is two right angles.

- The vertically opposite angles formed by the intersection of two straight lines are equal.

- Axiom 1 – Quantities which are equal to the same quantity are equal.
- Axiom 2 – Quantities which are obtained by adding equal quantities to equal quantities are equal.
- Axiom 3 – Quantities which are obtained by subtracting equal quantities from equal quantities are equal.
- Axiom 4 – Products which are equal quantities multiplied by equals are equal.
- Axiom 5 – Quotients which are equal quantities divided by nonzero equals are equal.

- If the two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent ( S.A.S )

- If two angles and a side of one triangle are equal to two angles & the corresponding side of another triangle, then the two triangles are congruent. ( A.A.S )

- If the three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent. ( S.S.S )

- If the lengths of the hypotenuse & a side of a right. angled triangle are equal to the lengths of the hypotenuse & a side of another right-angled triangle, then the two triangles are congruent ( R.H.S )

- The exterior angle formed when a side of a triangle is produced is equal to the sum of the two interior opposite angles.

- The sum of the interior angles of a triangle is 180°.

- If two sides are equal in a triangle, the angles opposite the equal sides are equal.

- In an isosceles triangle,
- i) the perpendicular is drawn from the apex to the opposite side.
- ii) the bisector of the apex angle.
- iii) the straight line joining the apex to the midpoint of the opposite side.
- iv) The perpendicular bisectors of the side opposite the apex, coincide with each other.

- The converse of isosceles triangles,

- The sides opposite equal angles of a triangle are equal.

- The opposite sides of a parallelogram are equal, the opposite angles of a parallelogram are equal, & the area of the parallelogram is bisected by each diagonal.

- In a parallelogram,
- i) opposite sides are equal.
- ii) opposite angles are equal.
- iii) the area of the parallelogram is bisected by each diagonal.

- In a parallelogram, the diagonals bisect each other.

- If the opposite sides of a quadrilateral are equal, then it is a parallelogram.

- If the opposite angles of a quadrilateral are equal, then it is a parallelogram.

- If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

- In a quadrilateral, if a pair of opposite sides is equal & parallel, then the quadrilateral is a parallelogram.

- The straight line joining the center of the circle to the midpoint of a chord is perpendicular to the chord.

- The perpendicular drawn from the center of a circle of a chord bisects the chord.

- The locus of a point moving at a constant distance from a fixed point is a circle.

- The locus of a point moving at an equal distance from two fixed points is the perpendicular bisector of the straight line joining the two points.

- The locus of a point moving at a constant distance from a given straight line is a line parallel to the given straight line, at the given constant distance from the straight line, which may lie on either side of the straight line.

- The locus of a point moving at an equal distance from two intersecting straight lines is a bisector of the angle formed at the intersection point of the lines.

- Two angles subtended by one are at the center if a circle is twice the angle subtended by the same area on the remaining part of the circle.

- The angles in the same segment of a circle are equal.

- An angle in a semicircle is a right angle.

- Parallelograms on the same base and between the same pair of parallel lines are equal in area.

- If a triangle and a Parallelogram lie on the same base and between the same pair of parallel lines, then the area of the triangle is exactly half the area of the Parallelogram.

- Triangles on the same base and between the same pair of parallel lines are equal in area.

- The straight line segment through the midpoints of two sides of a triangle is parallel to the third side and equal in length to half of it.

- The straight line through the midpoint of one side of a triangle and parallel to another side bisects the third side.

- A line drawn parallel to a side of a triangle divides the other two sides proportionally.

- If a line divides two sides of a triangle proportionally, then that line is parallel to the other side.

- If two triangles are equiangular then the corresponding sides are proportional.

- If the three sides of a triangle are proportional to the three sides of another triangle, then the two triangles are equiangular.

- In a right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the remaining side of the triangle, which includes the right angle.

- The opposite angles of a cyclic quadrilateral are supplementary.

- If the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral are on the circle.

- If one side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle of the quadrilateral.

- The straight line drawn through a point on a circle and perpendicular to the radius through the point of contact is a tangent to the circle converse.

- The tangent through a point on a circle is perpendicular to the radius drawn to the point of contact.

- If two tangents are drawn to a circle from an external point, then,
- i) the two tangents are equal in length.
- ii) the angle between the tangents is bisected by the straight line joining the external point to the center.
- iii) the tangents subtend equal angles at the center.

- The angles which a tangent to a circle makes with a chord drawn from the point of contact are respectively equal to the angles in the alternate segments of the circle.

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