# Kerala Syllabus 6th Standard Maths Solutions Chapter 8 Joining Angles

## Kerala State Syllabus 6th Standard Maths Solutions Chapter 8 Joining Angles

### Joining Angles Text Book Questions and Answers

Joining Angles Textbook Page No. 123

Your geometry box has two set squares. Each of them has three angles.

Look at an angle drawn with a corner of a set square.

How much is ∠CAB?
What if we draw another angle on top of this, using a corner of the other set square?
∠CAB = 30
Explanation:
By observing the figure triangle ABC, we came to conclude that,
as ∠CAB is 30 degrees, ∠ABC is 90 degrees and angle ∠BCA is 60 degrees
These set squares come in two usual forms,
both right triangles one with 90-45-45 degree angles,
the other with 30-60-90 degree angles.
Combining the two forms by placing the hypotenuses together will get 15° and 75° angles.

How much is ∠DAC?
And ∠DAB?
75°

Explanation:
By observing set square, the figure angle ∠DAB,
we conclude that, as ∠DAB is 75 degrees, ∠CAB is 30 degrees and angle ∠DAC is 45 degrees.
∠DAC = 45° And ∠DAB = 75°

∠DAB = 75°

Explanation:
By observing set square, the figure angle ∠DAB,
we conclude that, as ∠DAB is 75 degrees, ∠CAB is 30 degrees and angle ∠DAC is 45 degrees.
∠DAC = 45° And ∠DAB = 75°

Now suppose we draw angles as shown below:

How much ∠DAC?
Like this, angles of what different measures can we draw using the two set squares?
15°
Explanation:
These set squares come in two usual forms, both right triangles
one with 90-45-45 degree angles, the other with 30-60-90 degree angles.
Combining the two forms by placing the hypotenuses together will get 15° and 75° angles.

In each picture below, two angles are given. Calculate the third angle as a sum or difference;

∠DAC = ____ – ____ = _____
40°
Explanation:
∠DAC = 40°
∠CAB = 60°
∠DAB = 40° + 60° = 100°

∠DAB = ____ – ____ = _____
75°
Explanation:
∠DAB = 30° + 60° = 90°

∠DAC = ____ – ____ = _____
50°
Explanation:
Given;
∠DAC = 80°; ∠CAB = 30°
∠DAC = 80° – 30° = 50°

∠DAC = ____ – ____ = _____
75°
Explanation:
Given;
∠DAB = 120°; ∠CAB = 45°
∠DAC = 120° – 45° = 75°

Two sides Textbook Page No. 126

Draw a line and then a perpendicular at one end.

We have noted that such an angle measures 90°.
Now extend the horizontal line a bit to the left;

Now there is another angle on the left of the vertical line also. What is its measure?
Perpendicular means straight up, not leaning to the left or right.
So the angle on the left is also 90°.

Now let’s draw a slanted line through the foot of the perpendicular.

How much is the angle on the left of this slanted line’?
A bit more than 90°, right’?
How much more’?

By how much is the angle on the right less than 90°?

Now can’t we calculate the angle on the left also?

see this picture

How much is the angle on the left of the slanted line? Imagine a perpendicular through the point where the lines meet.
By how much is the angle on the right less than 90°?
So, by how much is the angle on the left more than 90°?
Thus the angle on the left is 90° + 40° = 130°.

Joining setsquares Textbook Page No. 128

The picture shows two identical setsquares placed together;

What are the measures of the angles of this triangle?
60°
Explanation:
The sum of the equilateral triangle is 120°
60° + 60° + 60° = 120°
60° in three corners as shown in the below figure

In the pictures below, two angles are marked and one of them is given. Find the other:

80°
Explanation:

As we know a straight line is having 180° angle.
∠AOB = 180°
∠AOC = ∠AOB – ∠COB
∠AOC = 180° – 100° = 80°

140°
Explanation:

As we know a straight line is having 180° angle.
∠AOB = 180°
∠AOC = ∠AOB – ∠COB
∠AOC = 180° – 40° = 140°

130°
Explanation:

as we know a straight line is having 180° angle
∠AOB = 180°
∠AOC = ∠AOB – ∠COB
∠AOC = 180° – 50° = 130°

100°
Explanation:

As we know a straight line is having 180° angle
∠AOB = 180°
∠AOC = ∠AOB – ∠COB
∠AOC = 180° – 80° = 100°

Meeting lines

See these pictures:

All of them show two lines meeting ; and each has two angles, one on the left and one on the right.
In the first picture, both angles are 90°. In the second ,the angle on the right is less than 90° and the angle on the left is greater than 90°; in the third picture , it is the other way round.

In the second and third picture, the angle on one side is that much more than 90° as the angle on the other side is less than 90°.
So, the sum of the angles on either side is 90° + 90° = 180°, right?

Thus we can write this as a general principle:
When two lines meet, the sum of the angles on either side is 180°.

Two such angles, made by two lines meeting is called a liner pair. So we can state this principle like this also:
The sum of the angles in a linear pair is 180°.

Textbook Page No. 130

Question 1.
How much is ∠ACE in the picture below?

105°
Explanation:
as we know a straight line is having ∠ACB 180° angle
∠ACB = 180°
∠ACE = ∠ACB – ∠ECB
∠ACE = 180° – 75° = 105°

Question 2.
What is the angle between the lines in this picture?

75°
Explanation:
Sum of the angles = 180°
with the help of a protractor we can measure the angles as 45° and 60°
180° – (45° + 60°)
180° – 105° = 75°

Question 3.
In the picture below, ∠ACE = ∠BCD. Find the measure of each.

150°
Explanation:
Sum of the angles = 180°
∠ACE + ∠BCD = 180° – ∠DCE
180° – 30° = 150°
∠ACE + ∠BCD = 150°
∠ACE = 75°
∠ACE = ∠BCD = 75°

Question 4.
One angle of a linear pair is twice the other. How much is each?
60° and 120°
Explanation:
Sum of the angles = 180°
If one angle of linear is 60°
twice the other is 60° + 60° = 120°

Question 5.
The angles in a linear pair are consecutive odd numbers. How much is each’?
89° and 91°
Explanation
A linear pair are consecutive odd number.
x + x + 2 = 180°
2x + 1 = 180°
2x = 178°
x = 89°
The two consecutive odd numbers are 89° and and 91°

So, required two angles that are consecutive odd numbers and are in a linear pair are 89° and 91°.

Crossing lines

See the picture:

How much is the angle on the left?
What if we extend the top line downwards crossing the horizontal line?

Now we have two angles below also. What are their measures?
The angles above and below, on the right of the slanted line, form a linear pair, right’?

Thus don’t we get one angle below?
Like this, the angle above and below on the left also form a linear pair.

Thus we get the angle below on the left also. Let’s look at all the angles together:

Linear pair Textbook Page No. 131

Draw the line AB and a point C on it. Draw a circle centered at C. Mark a point D on the circle.

Join CD. Now we can hide the circle. By choosing Angle and clicking on B, C, D in order, we get the measure of ∠BCD. In the same way, click on A, C, D to get ∠ACD.

Using Move change the position of D. How do the angles change? Look at the sum of BCD and ACD.
Yes,
Explanation:
There is no change in the angles after the move position of D.
The sum of the angle ∠BCD + ∠ACD = 180° degrees

Textbook Page No. 132

Some pictures showing two lines crossing each other are given below. One of the four angles so formed is given. Calculate the other three and write them in the pictures.

The angles three are 135°, 45°, 135° as shown in the below figure,

Explanation:
∠AOB is straight angle is equal to 180°
∠BOD and ∠AOD  are adjacent angles,
the sum of ∠BOD and ∠AOD is 180°
∠BOD = 45°
∠AOD = 180° – 45°
∠AOD = 135°
As opposite angle of a bisecting line at one center point is equal.
∠COB and ∠AOD are equal to 135°
∠BOD and ∠AOC are equal to 45°

The three angles are 60°, 120°, 60° as shown in the below figure,

Explanation:
∠AOB is straight angle is equal to 180°
∠BOD and ∠AOD  are adjacent angles,
the sum of ∠BOD and ∠AOD is 180°
∠AOD = 120°
∠BOD = 180° – 120°
∠BOD = 60°
as opposite angle of a bisecting line at one center point is equal.
∠COB and ∠AOD are equal to 120°
∠BOD and ∠AOC are equal to 60°

90 degrees as shown below,

Explanation:
∠AOB is straight angle is equal to 180°
∠BOD and ∠AOD  are adjacent angles and the sum of ∠BOD and ∠AOD is 180° degrees
∠BOD = 90°
∠AOD = 180° – 90°
∠AOD = 90°
as opposite angle of a bisecting line at one center point is equal
∠COB and ∠AOD are equal to 90°
∠BOD and ∠AOC are equal to 90°

Near and opposite

The picture shows the four angles made by the line CD crossing the line AB:

We can pair these four angles in various ways. Of these, four are linear pairs. which are they?

• ∠APC
• ∠BPC
•∠APD
•∠DPB
These are nearby angles in the picture. What about the other two pairs?

•∠APC, ∠BPD are opposite angles
•∠APC, ∠BPC are opposite angles

• ∠APC, ∠BPD
• ∠APD, ∠BPC

They are not nearby angles; they are opposite angles. What is the relation between them?

Look at ∠APC and ∠BPD. If we add ∠BPC to any of these, we get 180°. In other words, each of these is ∠BPC subtracted from 180°.
So, ∠APC = ∠BPD.
Similarly, can’t you see that the other pair of opposite angles are also equal?

This we write as a general principle;
The opposite angles formed by two lines crossing each other are equal.

We can combine the general result on nearby and opposite angles.
Of the four angles formed by two lines crossing each other, the sum of the nearby angles is 1800, the opposite angles are equal.

Draw a circle center at a point A. Mark four points B, C, D, E on the circle.

Draw the lines BD and CE. Now hide the circle.
Use Angle to mark the four angles in the picture.
Use Move to change of B, C, D, E.
Observe what happens to the opposite angles.

In the picture above the sum of the green and red angles is 180° and the sum of the green and blue angles is also 180°. So the red and blue angles are equal. Can you see that the green and yellow angles are equal?
Yes, the red and blue angles are equal.
Explanation:
As opposite angles of a bisecting two lines at one point, the opposite angles are equal.
So, the green and yellow angles are equal.
Angles that are opposite to each other when two lines cross, are also known as vertical angles, because the two angles share the same corner.
Opposite angles are also congruent angles, when they are equal or have the same measurement.

Question 1.
Two picture of lines passing through a point are given below. Some of the angles are given. Calculate the other angles marked and write in the figure:

Explanation:
As A and A’ and X and X’ are straight lines and the straight angle is 180 degrees.
With reference to the center point and bisecting lines of BB’ and CC’ angles are noted.
Same as in the figure XX’ bisects the lines YY’ and ZZ’ at common center the angles are noted. Opposite angles are opposite to each other when two lines cross, are also called vertical angles, because the two angles share the same corner.
Opposite angles are also congruent angles, because they are equal or have the same measurement.

Question 2.
Of the four angles made by two lines crossing each other, one angle is half of another angle. Calculate all four angles.
The four angles are 60°, 120°, 60°, 120°.
Explanation:
Linear pair,
∠a + ∠b = 180°.
∠c + ∠d = 180°.
∠a = ∠c
∠b = ∠d
Let ∠a = 2x
∠b = half of ∠a
∠b = x
∠a + ∠b = 180°.
2x + x = 180°.
3x = 180°.
x = 180/3
x = 60°.
∠b = 60°, ∠c = 60°.
∠a = 2x
∠a = 2 x 60
∠a = 120°.
So, the four angles are 60°, 120°, 60°, 120°.

Question 3.
Of the four angles formed by two lines crossing each other, the sum of two angles is 100°. Calculate all four angles.
Four angles are 50°, 130°, 50°, 130°
Explanation:
Linear pairs lie on the same line.
Sum of the four angles = 180°.
∠A + ∠C = 100°.
x + x = 100°.
2x = 100°.
x = 100/2 = 50°.
Linear pair,
∠A + ∠B = 180°.
50° + ∠B = 180°.
∠B = 180°- 50°.
∠B = 130°
Linear pair,
∠C + ∠D = 180°.
50° + ∠D = 180°.
∠D = 180°- 50°.
∠D = 130°
So, the four angles are 50°, 130°, 50°, 130°