Hello friends, in today’s session we are going to learn about the Symmetry and congruence of geometric figures. We will also study how we can use reflection to verify symmetry. But first we should know the Algebra Answers of what geometry is and what geometrical figures are.
Geometry is a branch of mathematics which is concerned with shape, size and relative position of figures. Geometry is one of the oldest mathematical sciences. There are four types of geometries Euclidean Geometry, Differential Geometry, Topology and the Algebraic geometry
In geometry what we are concerned is mainly the different shapes. Ant point, line, segment, ray, angle, polygon, curve, region, plane, surface, solid etc. are the geometric figures or in other words we can say that a geometric figure is any set of points either in a plane or in space.
A polygon is a closed figure made by connecting line segments, where each line segment end connects to only one end of two other line segments.
A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, angle is 180° × (n - 2) degrees.
A triangle is a three-sided polygon. The sum of the angles of a triangle is always equal to 180 degrees.
A triangle having all three sides of equal length is known as an equilateral triangle. All the angles of an equilateral triangle measure 60 degrees.
A triangle having two sides of equal length is known as an isosceles triangle.
A Scalene triangle is one having all the three sides of different lengths.
A four-sided polygon is known as a Quadrilateral. The sum of the angles of a quadrilateral is 360 degrees.
A four-sided polygon having all right angles is known as a rectangle. All the angles of a rectangle also sum up to 360 degrees.
A four-sided polygon having equal-length sides meeting at right angles is known as a square.
A four-sided polygon with two pairs of parallel sides is known as a parallelogram.
A four-sided polygon having all four sides of equal length and opposite sides parallel is known as a Rhombus.
A four-sided polygon having exactly one pair of parallel sides is said to be a trapezoid. The two sides that are parallel are called the bases of the trapezoid.
A five-sided polygon is known as a pentagon. The sum of the angles of a pentagon is 540 degrees.
Hexagon is a six-sided polygon. The sum of the angles of a hexagon is 720 degrees.
A seven-sided polygon is known as a Heptagon. The sum of the angles of a heptagon is 900 degrees.
An eight-sided polygon is known as an Octagon. The sum of the angles of an octagon is 1080 degrees.
A nine-sided polygon is known as a Nonagon. The sum of the angles of a nonagon is 1260 degrees.
A ten-sided polygon is known as a Decagon. The sum of the angles of a decagon is 1440 degrees.
A circle is the collection of points in a plane that are all the same distance from a fixed point. The fixed point is called the center. A line segment joining the center to any point on the circle is called a radius.
As we know what the figures are so let’s start with the symmetry again.
In easy language the symmetric figures are those they can be split in to exactly two halves. To understand it more let’s do it practically. Take a piece of colored paper and fold it into two equal halves, cut the paper in shape of a triangle from the middle. Leave the fold attached; now when we open the two ends of the heart we can see that the two parts are exactly the reflection of each other or we can say that both these parts are symmetrical in shape. The center line that divides it into two equal parts is known as the line of symmetry.
There are different types of symmetry-
1. Mirror symmetry- it is the same as we have learned above, i.e. when two parts are exactly the mirror image of each other then they are send to have a mirror symmetry. This could be understood by seeing the example given below. In which the figure is in the S plane and A is the line of symmetry.
2. Central symmetry – a figure is said to be central symmetrical if it has symmetry about the center doesn’t matter that they are mirror images or not. We can see in the figure given below that it is symmetrical about the point C, as it divides the line AE in two equal parts. And the triangles CDE and CAB are identical in shape.
3. Rotation Symmetry - A body is said to have rotation symmetry, if turning by an angle 360 °/n around some straight line AB which is the symmetric axis it coincides completely with its initial position.
For Example – in this figure we have axial symmetry.
By learning this you might be able to understand what the different kinds of symmetry are, but you might still have a confusion that how to distinguish between them. Let’s understand this with the help of examples which have all the three symmetries.
For Example- A sphere has all the three kinds of symmetry. The center symmetry is at the center of the ball, a plane or mirror symmetry is by a plane of any large circle, a symmetry axis is by the diameter of the circle. For axial symmetry we can take the round cone, which has the axis of the cone as its symmetry axis.
So this is all about the symmetry of geometric figures. Now we move over to the congruence of the geometric figures.
Congruent figures are those if they have the same shape and size. It means that if we reposition our one figure then it could coincide it precisely with another figure, it can also be said that two figures are congruent if one can be exactly transformed into another by isometry. The congruence could be either of triangles, angles etc.
The property of congruency is generally used for solving problems based on triangles. Two triangles are said to be congruent if they have same shape and size. To elaborate it the triangles are said to be congruent if the sides and angles of one triangle are equal to the sides and angles of the other one than they are said to be congruent. Like For example if the two triangles are congruent then they can be mathematically written as ∠ABC ≅ ∠PQR, where the sign ≅is used to represent two congruent figures. There are different ways to check whether a triangle is congruent or not.
So let’s see what these methods are
1. SAS (Side Angle Side) – if the two pairs of side in a triangle are equal in length and their included angle is also equal then they are said to be congruent.
2. SSS (Side Side Side) – if all the three pairs of side of two triangles are equal then they are said to be congruent.
3. ASA (Angle Side Angle) – if the two angles of two triangles are equal in measurement and their included side is also equal then they are said to be congruent.
4. AAS (Angle Angle Side) – if two angles in two triangles and the side not included is also equal then the triangles are also said to be congruent.
5. RHS (Right Angle hypotenuse Side) – if two right angled triangles have their Hypotenuse and shorter size equal in length then the triangles are said to be congruent.
If in a triangle any of these is satisfied then it is said to be a congruent triangle. Let’s understand this with the help of an example.
For SSS property.
For SAS property.
For ASA property.
For AAS property.
By RHS property.
Now as we know what symmetry and congruency are, so we move towards our next topic of the day that is Reflections to verify symmetry. As we have learned above that two figures are symmetrical if they have a mirror image about their line of symmetry.
How to test a relation for Symmetry?
When we graph a relation given by an equation we get some difficulties, but what if these graphs are symmetrical and we find out that, then it would become very easy for us.
These are some graphs which are symmetrical about Y axis.
For the special lines, the y axis and the x axis there is a very easy way to find the reflection of any point. For the y axis you just change the sign of the x and for the x axis you change the sign of the y. so by this we can find whether a graph is symmetrical with respect to the axis.
There are also figures that are symmetrical about the origin but they are bit difficult to find.
In upcoming posts we will discuss about Numbers in Grade IV and Tools to solve problems. Visit our website for information on CBSE psychology question paper 2010
Geometry is a branch of mathematics which is concerned with shape, size and relative position of figures. Geometry is one of the oldest mathematical sciences. There are four types of geometries Euclidean Geometry, Differential Geometry, Topology and the Algebraic geometry
In geometry what we are concerned is mainly the different shapes. Ant point, line, segment, ray, angle, polygon, curve, region, plane, surface, solid etc. are the geometric figures or in other words we can say that a geometric figure is any set of points either in a plane or in space.
A polygon is a closed figure made by connecting line segments, where each line segment end connects to only one end of two other line segments.
A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, angle is 180° × (n - 2) degrees.
A triangle is a three-sided polygon. The sum of the angles of a triangle is always equal to 180 degrees.
A triangle having all three sides of equal length is known as an equilateral triangle. All the angles of an equilateral triangle measure 60 degrees.
A triangle having two sides of equal length is known as an isosceles triangle.
A Scalene triangle is one having all the three sides of different lengths.
A four-sided polygon is known as a Quadrilateral. The sum of the angles of a quadrilateral is 360 degrees.
A four-sided polygon having all right angles is known as a rectangle. All the angles of a rectangle also sum up to 360 degrees.
A four-sided polygon having equal-length sides meeting at right angles is known as a square.
A four-sided polygon with two pairs of parallel sides is known as a parallelogram.
A four-sided polygon having all four sides of equal length and opposite sides parallel is known as a Rhombus.
A four-sided polygon having exactly one pair of parallel sides is said to be a trapezoid. The two sides that are parallel are called the bases of the trapezoid.
A five-sided polygon is known as a pentagon. The sum of the angles of a pentagon is 540 degrees.
Hexagon is a six-sided polygon. The sum of the angles of a hexagon is 720 degrees.
A seven-sided polygon is known as a Heptagon. The sum of the angles of a heptagon is 900 degrees.
An eight-sided polygon is known as an Octagon. The sum of the angles of an octagon is 1080 degrees.
A nine-sided polygon is known as a Nonagon. The sum of the angles of a nonagon is 1260 degrees.
A ten-sided polygon is known as a Decagon. The sum of the angles of a decagon is 1440 degrees.
A circle is the collection of points in a plane that are all the same distance from a fixed point. The fixed point is called the center. A line segment joining the center to any point on the circle is called a radius.
As we know what the figures are so let’s start with the symmetry again.
In easy language the symmetric figures are those they can be split in to exactly two halves. To understand it more let’s do it practically. Take a piece of colored paper and fold it into two equal halves, cut the paper in shape of a triangle from the middle. Leave the fold attached; now when we open the two ends of the heart we can see that the two parts are exactly the reflection of each other or we can say that both these parts are symmetrical in shape. The center line that divides it into two equal parts is known as the line of symmetry.
There are different types of symmetry-
1. Mirror symmetry- it is the same as we have learned above, i.e. when two parts are exactly the mirror image of each other then they are send to have a mirror symmetry. This could be understood by seeing the example given below. In which the figure is in the S plane and A is the line of symmetry.
2. Central symmetry – a figure is said to be central symmetrical if it has symmetry about the center doesn’t matter that they are mirror images or not. We can see in the figure given below that it is symmetrical about the point C, as it divides the line AE in two equal parts. And the triangles CDE and CAB are identical in shape.
3. Rotation Symmetry - A body is said to have rotation symmetry, if turning by an angle 360 °/n around some straight line AB which is the symmetric axis it coincides completely with its initial position.
For Example – in this figure we have axial symmetry.
By learning this you might be able to understand what the different kinds of symmetry are, but you might still have a confusion that how to distinguish between them. Let’s understand this with the help of examples which have all the three symmetries.
For Example- A sphere has all the three kinds of symmetry. The center symmetry is at the center of the ball, a plane or mirror symmetry is by a plane of any large circle, a symmetry axis is by the diameter of the circle. For axial symmetry we can take the round cone, which has the axis of the cone as its symmetry axis.
So this is all about the symmetry of geometric figures. Now we move over to the congruence of the geometric figures.
Congruent figures are those if they have the same shape and size. It means that if we reposition our one figure then it could coincide it precisely with another figure, it can also be said that two figures are congruent if one can be exactly transformed into another by isometry. The congruence could be either of triangles, angles etc.
The property of congruency is generally used for solving problems based on triangles. Two triangles are said to be congruent if they have same shape and size. To elaborate it the triangles are said to be congruent if the sides and angles of one triangle are equal to the sides and angles of the other one than they are said to be congruent. Like For example if the two triangles are congruent then they can be mathematically written as ∠ABC ≅ ∠PQR, where the sign ≅is used to represent two congruent figures. There are different ways to check whether a triangle is congruent or not.
So let’s see what these methods are
1. SAS (Side Angle Side) – if the two pairs of side in a triangle are equal in length and their included angle is also equal then they are said to be congruent.
2. SSS (Side Side Side) – if all the three pairs of side of two triangles are equal then they are said to be congruent.
3. ASA (Angle Side Angle) – if the two angles of two triangles are equal in measurement and their included side is also equal then they are said to be congruent.
4. AAS (Angle Angle Side) – if two angles in two triangles and the side not included is also equal then the triangles are also said to be congruent.
5. RHS (Right Angle hypotenuse Side) – if two right angled triangles have their Hypotenuse and shorter size equal in length then the triangles are said to be congruent.
If in a triangle any of these is satisfied then it is said to be a congruent triangle. Let’s understand this with the help of an example.
For SSS property.
For SAS property.
For ASA property.
For AAS property.
By RHS property.
Now as we know what symmetry and congruency are, so we move towards our next topic of the day that is Reflections to verify symmetry. As we have learned above that two figures are symmetrical if they have a mirror image about their line of symmetry.
How to test a relation for Symmetry?
When we graph a relation given by an equation we get some difficulties, but what if these graphs are symmetrical and we find out that, then it would become very easy for us.
These are some graphs which are symmetrical about Y axis.
For the special lines, the y axis and the x axis there is a very easy way to find the reflection of any point. For the y axis you just change the sign of the x and for the x axis you change the sign of the y. so by this we can find whether a graph is symmetrical with respect to the axis.
There are also figures that are symmetrical about the origin but they are bit difficult to find.
In upcoming posts we will discuss about Numbers in Grade IV and Tools to solve problems. Visit our website for information on CBSE psychology question paper 2010
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