Angles
Recall an angle and recognize acute, right, obtuse, straight and reflex angle
We have learn’t about an angle and its different types in previous classes. However we recall these concepts
Angle
We have learnt in grade 4 that an angle is formed by two distinct rays with the same endpoint. The common endpoint is called the vertex. The figure to the right is of an angle O, AOB or BOA. The symbol for an angle is ∠.
Acute Angle
The angle ABC given on the right side is an acute angle because its measure is less than 90° i.e; ∠ABC <90°
Right Angle
∠ABC is a right angle because the measure of ∠ABC is equal to 90°. i.e. ∠ABC = 90°
Obtuse Angle
∠LMN is an obtuse angle because its measure 120° is greater than 90° and less than 180° i.e. ∠LMN>90°
Straight Angle
∠PQR is a straight angle which is formed by two adjacent right angles PQS and SQR.
∠PQR = ∠PQS + ∠SQR = 90 + 90 = 180°
It is clear from the figure that \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\) are two rays in opposite directions with common point Q as vertex.
Reflex Angle
AOB is a reflex angle because its measure is 225° which is greater than 180° and less than 360°.
Draw Acute and Obtuse angles of different measures using Protractor
Draw an Acute Angle
Draw an angle of measure 40°.
Steps of Construction:
(I)Draw a ray QR
(II)Place the straight edge of the protractor such that its central point falls on and the ray joining the central point to the mark 0 coincides with ray QR.
(III)Read the protractor from the inner side where its mark lies on the ray QR.
(IV)Mark a point P near the circular edge make 40º as given in the figure.
(V)Remove the protractor and draw the ray QP as shown in the picture.
(VI)Thus ∠PQR = 40º is the required acute angle.
Draw an Obtuse Angle
Draw an angle of measure 130°.
Steps of Construction
(I)Draw a ray MN.
(II)Place the straight edge of the protractor such that its central point O falls on M and the ray joining the central point to the mark O coincides with ray MN.
(III)Read the protractor from the inner scale where its zero (0) mark lies on the ray MN.
(IV)Mark a point L near the edge marked 130° as shown in the figure
(V)Remove the protractor and draw the ray as shown in the adjoining figure.
(VI)Thus, ∠LMN = 130° is the required obtuse angle
Draw an angle equal in measure to a given angle
Steps of Construction
(I)Measure the given angle ABC with the help of a protractor that is ∠ABC=50°. We have to draw an angle equal in measure of given angle i.e. 50°. We proceed further as under.
(II) Draw a ray QR with as the initial point (vertex)
(iii) Place the centre of the protractor on Q and adjust it such that its straight edge or base line coincide with ray QR.
(iv) Start from zero (0) and read the inner scale till we reach the mark 50.
(v) Mark a point P against the mark 50.
(vi)Remove the protractor and draw the ray QP.
(vii) Thus ∠PQR which is the required angle equal in measure to the given angle.
Draw an angle twice in measure to a given angle.
Steps of Construction
(i)Measure the given angle ABC with the help of a protractor ∠ABC=40°.
We have to draw an angle twice in measure of the given angle i.e the measure will be 2 x 40°= 80° To draw an angle of measure 80° with the help of a protractor, we proceed further as below.
(ii) Draw a ray QR with as the initial point.
(iii) Place the centre of the protractor on such that its baseline coincide with ray QR.
(iv) Start from 0 and read the inner scale till we reach the mark 80.
(v) Mark a point P against the mark 80
(vi) Remove the protractor and draw the ray QP
Thus, ∠PQR=80° is the required angle twice in measure to the given angle ABC.
Draw an angle equal in measure to the sum of two angles
Steps of Construction
(I)Measure the given angles ABC and LMN with the help of protractor and note that ∠ABC = 40° and ∠LMN = 80° The sum of measures of the given angles is 40° + 80° = 120° We have to draw an angle equal in measure to the sum of measures two given angles i.e, 120°. We proceed further as below.
(ii) Draw a ray QR with as the initial point.
(iii) Place the centre of the protractor on such that its baseline coincides with ray QR
(iv) Start from zero and read the inner scale till we reach at the mark 120°.
(v) Mark a point P near the mark 120°.
(vi) Remove the protractor and draw the ray QP.
Thus, ∠PQR = 120° is the required angle equal in measure to the sum of two given angles.
Construction of Angels
We have to construct a right angle, a straight angle and a reflex angle.
We shall construct these angles one by one.
Right Angle
Construct an angle whose measure is 90°
Steps of construction:
(i) Draw a ray MN.
(II) Place a protractor on MN such that its central point O falls on M and the ray joining the central point to the mark zero coincides with the ray MN.
(iii) Read the protractor from the inner side where zero mark lies on the ray MN till we reach the mark 90°
(iv) Mark a point L near the mark 90° as shown in the figure.
(v) Remove the protractor and draw the ray ML as given in the figure
Thus, ∠LMN=90° is the required right angle.
Straight Angle
Straight Angle Construct an angle of measure 180°
Steps of Construction:
1.Draw a ray QR
2.Place a protractor on QR such that its central point falls on Q and the ray joining the central point to the mark zero coincides with the ray QR.
3.Read the protractor from the inner side where zero mark lies on the ray QR till we reach the mark 180°. Mark a point near the mark 180° as shown in the figure.
4.Remove the protractor and draw the ray QP as given in the figure
Thus, ∠PQR = 180° is the required straight angle.
Reflex Angle
Construct an angle of measure 210°. Now, 210° = 180° + 30°
Steps of Construction:
(i)Draw a ray YZ.
(ii)Place a protractor on \(\overrightarrow{YZ}\) such that its central point to fall on Y and the ray joining the central point to mark zero(0) coincides with the ray YZ.
(iii)Read the protractor from the outer side where its zero mark lies on the ray YZ till the mark 30°
(iv)Mark a point X near the mark 30° as given in the figure. The angle will become 180° + 30° = 210°
(V)Remove the protractor and draw the ray YX as shown in the figure.
Thus, ∠XYZ = 210° is the required reflex angle.
Reflex angles of different measures can be drawn in the same manner.
Triangles
Definition of a Triangle A triangle is a simple closed figure having three sides and three angles.
(i) In the given triangle ABC:
(ii)A, B and C are the vertices.
(iii)AB, BC and CA are the three sides.
Three angles are ∠ABC, ∠BCA and ∠BAC.
The symbol used for a triangle is ∆. So ∆ABC means triangle ∆ABC.
The triangle can be written in any one of six ways as ∆ABC, ∆CBA, ∆BAC, ∆CAB, ∆BCA and ∆ACB.
It may be noted that the order of the vertices does not matter while writing the name of a triangle
Definition of triangle with respect to their sides
(i)Equilateral Triangle
An equilateral triangle is a triangle in which all the three sides are equal in length
The triangle given on the right side is an equilateral triangle because its all three sides are equal in length.
i.e. \(\overline{AB}=\overline{BC}=\overline{CA}\)
(ii) Isosceles Triangle
An isosceles triangle is a triangle in which any two sides are equal in length
The figure on the right side is an isosceles triangle ABC because its two sides are equal in length.
i.e. \(\overline{AB}=\overline{AC}\)
(iii) Scalene Triangle
A scalene triangle is a triangle in which all the sides are of different lengths.
PQ, QR and PR are not equal.
Definition of triangles with respect to their Angles
(i) Acute Angled Triangle
An acute angled triangle is a triangle with all three angles are acute angles (less than 90º).
In the figure on the right side is an acute angled triangle because its all three angles are acute.
(ii) Obtuse Angled Triangle
An obtuse angled triangle is a triangle with one obtuse angle (greater than 90°).
ΔABC is an obtuse angled triangle because its one angle is obtuse angle i.e. ∠B = 120° (greater than 90%). We know that no triangle can have more than one obtuse angle because a triangle must have the sum of all three angles as 180°.
(iii) Right Angled Triangle
A right angled triangle is a triangle in which one angle is 90°. In the given figure ΔABC is a right angled triangle because its one angle B is a right angle i.e. ∠B = 90°.
Construction of triangles when three sides are given
Equilateral Triangle
Example:
Draw an equilateral triangle PQR whose measure of each side is 3cm.
Solution:
Steps of Construction
(i) Draw a line segment PQ = 3cm.
(ii)Taking P as centre draw an arc of radius 3cm over \(\overline{PQ}\)
(iiI) Taking Q as centre, draw an arc of radius 3cm over \(\overline{PQ}\) which cuts the first arc at point R
(iv) Join R with P and Q one by one.
Thus, ΔPQR is the required equilateral triangle.
(ii) Isosceles Triangle
Example:
Draw an Isosceles triangle LMN with measure of its two sides as 3cm each and measure of third side is 4cm.
Solution:
Steps of Construction:
(i)Draw a line segment LM such that \(\overline{LM}\) = 4cm.
(ii)Taking L as centre, draw an arc of radius 3cm over \(\overline{LM}\).
(iii) Taking M as centre, draw another arc of radius 3cm over \(\overline{LM}\), which cuts the first arc at point N.
(iv)Join N with L and M one by one.
Thus, ΔLMN is the required isosceles triangle.
Scalene Triangle
Example:
Draw an scalene triangle ABC with measure of its sides as AB = 4.5cm, BC = 4cm and AC = 3cm.
Solution:
Steps of Construction:
(i)Draw a line segment AB = 4.5 cm.
(ii) Taking A as centre, draw an arc of radius 3cm over \(\overline{AB}\).
(iii) Taking B as centre, draw another arc of radius 4cm over \(\overline{AB}\), which cuts the previous arc at point C.
(iv) Join C with A and B one by one.
Thus, ∆ABC is the required scalene triangle.
Quadrilateral
A closed plane figure with four sides is known at as a quadrilateral. It has also four angles and four vertices.
Recognize the kinds of Quadrilateral
Following are the different kinds of quadrilateral.
(i) Square (ii) Rectangle (iii) Kite
(iv) Parallelogram (v) Rhombus (vi) Trapezium
Construction of Square and Rectangle
Square
We know that a square has four equal sides and each angle is of 90° .
Example
Construct a square with length of each side 2.5cm.
Solution
Steps of Construction:
(i)Draw a line segment LM = 2.5cm.
(ii)Construct an angle of 90° with the help of a protractor at the point L and at the point M.
(iii) Taking L as centre, draw an arc of radius 2.5 cm 2.5cm which cuts the vertical \(\overrightarrow{LP}\) at point O.
(iv)Taking M as centre, draw an arc of radius 2.5cm which cuts the vertical \(\overrightarrow{MQ}\) at point N.
(v) Join the point O with point N. Thus, LMNO is the required square
Rectangle.
In a rectangle each two opposite sides are equal in length and measure of each angle is of 90°. Construct a rectangle having length 4cm and width 3cm
Solution
Steps of Construction
(i)Draw a line segment AB = 4cm.
(ii)Construct an angle of 90° at point A with the help of a protractor.
(iii)Similarly draw an angle of 90° at point B.
(iv)Taking point A as centre, draw an arc of 3cm which cuts \(\overrightarrow{AF}\) at point D.
(v)Taking point B as centre, draw an arc of 3cm which cuts \(\overrightarrow{BE}\) at point C.
(vi)Join point C with point D. Thus, ABCD is the required rectangle