अध्याय 4: सरल समीकरण Class 7 Math

Line, line segment and ray

• If we take a point and draw a straight path that extends endlessly on both the sides, then the straight path is called as a line.
• A ray is a part of a line with one endpoint.
• A line segment is a part of a line with two endpoints.

Angles

• An angle is formed when two rays originate from the same end point.
• The rays making an angle are called the arms of the angle.
• The end point is called the vertex of the angle.

Complementary Angles

• Two angles whose sum is 90⁰ are called complementary angles.
  Example: 50⁰ + 40⁰ = 90⁰
  ∴ 50⁰ and 40⁰ angles are complementary angles.

Parallel Lines and a Transversal

Transversal intersecting two lines

• Transversal is a line that intersects two or more lines at different points.

• Corresponding Angles:
  (i) ∠1 and ∠5 (ii) ∠2 and ∠6
  (iii) ∠3 and ∠7 (iv) ∠4 and ∠8
• Alternate Interior Angles:
  (i) ∠3 and ∠6 (ii) ∠4 and ∠5
• Alternate Exterior Angles:
  (i) ∠1 and ∠8 (ii) ∠2 and ∠7
• Interior angles on the same side of the transversal:
  (i) ∠3 and ∠5 (ii) ∠4 and ∠6

Transversal of Parallel Lines

• If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
  (i) ∠1=∠5 (ii) ∠2=∠6
  (iii) ∠3=∠7 (iv) ∠4=∠8
• If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
  (i) ∠3=∠6 (ii) ∠4=∠5
• If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
  (i) ∠3+∠5=180⁰ (ii) ∠4+∠6=180⁰

Checking if two or more lines are parallel

• There are three conditions to check whether the two lines are parallel. They are:
  (i) If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.
  (ii) If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
  (iii) If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Intersecting Lines and Pairs of Angles

Supplementary angles

• Two angles whose sum is 180⁰ are called supplementary angles.
  Example: 110⁰+70⁰=90⁰
  ∴ 110⁰ and 70⁰ angles are supplementary angles.

Adjacent Angles

• Two angles are adjacent, if they have
  (i) A common vertex
  (ii) A common arm
  (iii) Their non-common arms on different sides of the common arm.

• Here ∠ABD and ∠DBC are adjacent angles.

Linear Pair

• Linear pair of angles are adjacent angles whose sum is equal to 180∘.

• Here, 1 and 2 are linear pair of angles.

Vertically Opposite Angles

• Vertically opposite angles are formed when two straight lines intersect each other at a common point.
• Vertically opposite angles are equal.

Here, the following pairs of angles are vertically opposite angles.
(i) a and c
(ii) b and d

Intersecting and Non-Intersecting lines

• Intersecting lines are lines which intersect at a common point called the point of intersection.

Parallel lines are lines which do not intersect at any point. Parallel lines are also known as non- intersecting lines.

Basic Properties of a Triangle

Sum of Interior Angles in a Triangle

• Angle sum property of a triangle: Sum of all interior angles of a triangle is 180⁰.

• In △ABC, ∠1+∠2+∠3=180⁰

The exterior angle of a triangle = Sum of opposite internal angles

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

In △ABC, ∠CAB+∠ABC=∠ACD.