Easy-to-read Ganita Prakash Class 7 Notes and Chapter 7 A Tale of Three Intersecting Lines Class 7 Notes save valuable study time during exam season.
Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines Notes
Class 7 A Tale of Three Intersecting Lines Notes
A triangle is a closed plane figure formed by three non-parallel line segments. The symbol “A” is used to denote a triangle (e.g., ∆ABC).
There are three types of triangles on the basis of the number of equal-length sides.
| Equilateral Triangle | Isosceles Triangle | Scalene Triangle |
| All three sides are equal in length. | Any two sides of a triangle are equal in length. | Three sides of a triangle are different in length. |
All three angles are equal, each measuring 60°. |
Angles opposite to equal sides are equal in measure. |
All three angles are of different measures and never equal. |
In a triangle, an altitude is a line segment drawn from a vertex perpendicular to its opposite side. In the given figure, AD, BE and CF are the altitudes of the ∆ABC.
Each triangle has 3 altitudes.
Triangle Inequality Theorem:
The sum of any two sides of a triangle is greater than the third side.
In the given triangle ABC, we have b + c > a, c + a > b and a + b > c.
Angle Sum Property: The sum of the three interior angles of any triangle is always 180°.
i.e., ∠A + ∠B + ∠C = 180°
Exterior Angle Property:
If a side of a triangle is extended, the exterior angle formed is equal to the sum of the two interior opposite angles.
i.e., ∠a + ∠b = ∠f, ∠b + ∠c = ∠d and ∠a + ∠c = ∠e
The sum of exterior angles of a triangle is 360°.
i.e., ∠d + ∠e + ∠f = 360°
There are three types of triangles on the basis of measure of their interior angles.
| Acute Angled Triangle | All three angles are less than 90° |  |
| Right Angled Triangle | One angle is 90° and other two are less than 90°. |  |
| Obtuse Angled Triangle | One angle is more than 90° and other two are less than 90°. |  |
To construct a triangle, we need at least three elements. These can be:
Three sides, or
Two sides and the included angle, or
Two angles and the included side
Construction of triangles when three sides are given
To construct a triangle ABC with sides AB = 6 cm, BC = 7 cm andTC = 8 cm, follow the steps laid down below:
Step 1: Draw a line segment AB of 6 cm using a ruler.
Step 2: With B as centre, draw7 an arc with radius 7 cm.
Step 3: With A as centre, draw an arc with radius 8 cm to cut the previous arc at point C.
Step 4: Join C to A and B. Thus, AABC is required triangle.
Construction of triangles when two sides and the included angle are given
If two sides (say AB = 5 cm and AC = 7 cm) along with the included angle (say ∠A then we follow the steps for constructing triangle as follows:
Step 1: Using a ruler, drawn a line segment AB of length 5 cm.
Step 2: Using protractor, draw the given vertex angle, ∠A = 50°.
Step 3: On the new arm created from ∠A, mark a point C such that AC = 7 cm using ruler and compass.
Step 4: Using a ruler, connect points B and C to complete the triangle ABC.
Construction of triangles when two angles and the included side are given
If two angles (say ∠A = 35° and ∠B = 60°) along with the included side (say AB = 6 cm) are given, then the steps for constructing triangle are as follows:
Step 1: Using a ruler, draw a line segment.-Id of length 6 cm.
Step 2: Using protractor, draw the given vertex angle, ∠A = 35°, by placing protractor at vertex A.
Step 3: Using protractor, draw the given vertex angle, ∠B = 60°, by accurately placing the protractor at vertex B.
Step 4: If the two new lines (arms of angles) are intersecting, then mark the intersection point as C. If not, then extend the two new lines such that they intersect at a point. Mark that point as C.
Thus, the required triangle, ∆ABC is constructed.
Triangle
A triangle is a dosed plane figuie ioimed by three non-parallel line segments. T he symbol “A” is used to denote a triangle (e.g., ∆ABC).
Altitude: In a triangle, an altitude is a line segment drawn from a vertex perpendicular to its opposite side. In the given figure, AD,
BE and CF are the altitudes of the ∆ABC.
NOTE:
Each triangle has 3 altitudes.
Two perpendicular sides in a right angled triangle are its altitudes.
Based on Angles
Acute Angled Triangle
All three angles are less than 90°.
Right Angled Triangle
One angle is 90° and other two are less than 90°.
Obtuse Angled Triangle
One angle is more than 90° and other two are less than 90°.
Based on Sides
Equilateral Triangle
All three sides are equal in length.
All three angles are equal, each measuring 60°.
Isosceles Triangle
Any two sides of a triangle are equal in length.
Angles opposite to equal sides are equal in measure.
Scalene Triangle
Three sides of a triangle are different in length.
