ICSE Class 6 Maths Chapter 18 Properties of Angles and Lines

Chapter 18

Properties Of Angles And Lines

(Including Parallel Lines)

Properties Of Adjacent And Vertically Opposite Angles

Property 1

When two straight lines intersect:

(i) the sum of each pair of adjacent angles is always 180°.

(ii) the vertically opposite angles are always equal.

For Example

In the adjoining figure, the straight lines AB and CD intersect each other at point O, and so, we have:

(i) the sum of adjacent angles = 180°.

i.e. \(\angle AOD + \angle DOB = 180°\), \(\angle BOD + \angle BOC = 180°\), \(\angle BOC + \angle COA = 180°\), and \(\angle COA + \angle AOD = 180°\).

(ii) the vertically opposite angles are equal, i.e. \(\angle AOC = \angle BOD\) and \(\angle BOC = \angle AOD\)

Property 2

If the sum of two adjacent angles is 180°, their exterior arms are always in the same straight line.

Conversely, if the exterior arms of two adjacent angles are in the same straight line, the sum of the angles is always 180°.

For Example

Considering the two adjacent angles given alongside:

(i) if \(\angle AOC + \angle BOC = 180°\), the exterior arms OA and OB are in the same straight line, i.e. AOB is a straight line.

(ii) if exterior arms OA and OB are in the same straight line, i.e. if AOB is a straight line, \(\angle AOC + \angle BOC = 180°\).

Example 1

Two straight lines AB and CD intersect at point P. If angle BPD = 54°, find, giving reason:

(i) \(\angle APD\)

(ii) \(\angle APC\).

Solution

(i) When two straight lines intersect each other, the adjacent angles are supplementary.

\(\Rightarrow \angle APD + \angle BPD = 180°\)

\(\Rightarrow \angle APD + 54° = 180°\)

\(\Rightarrow \angle APD = 180° - 54° = 126°\) (Ans.)

(ii) When two straight lines intersect each other, the vertically opposite angles are equal.

\(\Rightarrow \angle APC = \angle BPD = 54°\) (Ans.)

Example 2

The figure given alongside, shows two adjacent angles AOB and AOC whose exterior arms OB and OC are along the same straight line. Find the value of x.

Solution

Since the exterior arms of the adjacent angles are in a straight line, the adjacent angles are supplementary.

\(\therefore \angle AOC + \angle AOB = 180° \Rightarrow 2x + 10° + 70° = 180°\)

\(\Rightarrow 2x = 180° - 80° \Rightarrow x = \frac{100°}{2} = 50°\) (Ans.)

Example 3

Each figure given below shows a pair of adjacent angles AOB and BOC. Find whether or not the exterior arms OA and OC are in the same straight line.

Solution

(i) \(\angle AOB + \angle BOC = 90° + 70° = 160°\)

Since the sum of adjacent angles AOB and BOC is not 180°, the exterior arms OA and OB are not in the same straight line. (Ans.)

(ii) \(\angle AOB + \angle BOC = 95° + 85° = 180°\)

\(\Rightarrow\) The sum of adjacent angles AOB and BOC is 180°.

\(\therefore\) The exterior arms OA and OC are in the same straight line. (Ans.)

(iii) \(\angle AOB + \angle BOC = 90° + 105° = 195°\); which is not 180°.

\(\Rightarrow\) The exterior arms OA and OC are not in the same straight line. (Ans.)

Teacher's Note

Understanding how angles work when lines intersect helps us understand road intersections and how traffic lights determine safe crossing angles. When two roads cross, the angles created must follow these mathematical rules.

Exercise 18(A)

1. Two straight lines AB and CD intersect each other at a point O and angle AOC = 50°; find:

(i) angle BOD

(ii) \(\angle AOD\)

(iii) \(\angle BOC\)

2. The adjoining figure shows two straight lines AB and CD intersecting at point P. If \(\angle BPC = 4x - 5°\) and \(\angle APD = 3x + 15°\), find:

(i) the value of x.

(ii) \(\angle APD\)

(iii) \(\angle BPD\)

(iv) \(\angle BPC\).

3. The figure given alongside, shows two adjacent angles AOB and AOC whose exterior sides are along the same straight line. Find the value of x.

4. Each figure given below shows a pair of adjacent angles AOB and BOC. Find whether or not the exterior arms OA and OC are in the same straight line.

(i)

(ii)

(iii)

5. A line segment AP stands at point P of a straight line BC such that \(\angle APB = 5x - 40°\) and \(\angle APC = x + 10°\); find the values of x and angle APB.

Parallel Lines

Two straight lines are said to be parallel if they do not meet, no matter how much they be extended in either direction.

1. Two parallel lines are represented by drawing arrows on both lines in the same direction. See the figures drawn above.

2. The distance between two parallel lines does not change, i.e. neither does it increase nor does it decrease. That is why parallel lines never meet (intersect).

Teacher's Note

Parallel lines are everywhere in our daily lives - railway tracks, lane markings on highways, and the shelves in a bookstore all use the principle of parallel lines to stay organized and functional.

Concept Of Transversal Lines

When a line cuts two or more lines, whether (parallel or non-parallel), the line is called a transversal line, or simply, a transversal. In each of the following figures; PQ is a transversal line.

Angles Formed By Two Lines And Their Transversal Line

When a transversal cuts two parallel or non-parallel lines, eight (8) angles are formed: they are marked 1 to 8 in the following figure:

These angles can be distinguished as stated below:

1. Exterior Angles

The angles marked 1, 2, 7 and 8 are exterior angles.

2. Interior Angles

The angles marked 3, 4, 5 and 6 are interior angles.

3. Exterior Alternate Angles

The two pairs of exterior alternate angles are 2 and 8; 1 and 7.

4. Interior Alternate Angles

The two pairs of interior alternate angles are 3 and 5; 4 and 6. In general, interior alternate angles are simply called as alternate angles.

5. Corresponding Angles

The four pairs of corresponding angles are 1 and 5; 2 and, 6; 3 and 7; 4 and 8.

6. Co-interior Or Conjoined Or Allied Angles

The two pairs of co-interior or allied angles are 3 and 6; 4 and 5.

7. Exterior Allied Angles

The two pairs of exterior allied angles are 2 and, 7; 1 and 8.

When Two Parallel Lines Are Cut By A Transversal

1. Exterior alternate angles are equal, i.e. \(\angle 1 = \angle 7\) and \(\angle 2 = \angle 8\)

2. Interior alternate angles are equal, i.e. \(\angle 3 = \angle 5\) and \(\angle 4 = \angle 6\)

3. Corresponding angles are equal, i.e. \(\angle 1 = \angle 5\), and \(\angle 2 = \angle 6\); \(\angle 3 = \angle 7\) and \(\angle 4 = \angle 8\)

4. Co-interior (conjoined, allied) angles are supplementary, i.e. \(\angle 4 + \angle 5 = 180°\) and \(\angle 3 + \angle 6 = 180°\)

5. Exterior allied angles are supplementary, i.e. \(\angle 2 + \angle 7 = 180°\) and \(\angle 1 + \angle 8 = 180°\)

Example 4

In the figure given alongside, two parallel lines are cut by a transversal. Find, giving reasons, the values of the angles x, y and z.

Solution

\(\angle x = 80°\) [Vertically opposite angles]

\(\angle y = \angle x\) [Alternate angles]

\(= 80°\)

\(\angle x + \angle z = 180°\) [Co-interior angles are supplementary]

\(80° + \angle z = 180°\)

\(\Rightarrow \angle z = 180° - 80° = 100°\)

\(\therefore \angle x = 80°\), \(\angle y = 80°\) and \(\angle z = 100°\) (Ans.)

Teacher's Note

Understanding parallel lines and transversals is essential in architecture and construction - the framework of buildings, bridges, and roads all depend on getting these angles exactly right for safety and structural integrity.

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