Samacheer Kalvi Class 6 Maths Solutions Term 1 Chapter 4 Geometry Exercise 4.4

Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 1 Chapter 4 Geometry Ex 4.4

Miscellaneous Practise Problems

 

Question 1. Find the type of lines marked in thick lines (Parallel, intersecting or perpendicular)
Answer:
(i) These lines are parallel lines. They run side-by-side and never meet, like railroad tracks.
(ii) These are also parallel lines. They maintain the same distance apart, just like the opposite edges of a box.
(iii) This figure shows both parallel and perpendicular lines. The straight part of the roller is parallel to the ground, while the handle could be thought of as perpendicular if it's perfectly upright.
(iv) These are intersecting lines. They cross each other at a point, just like the blades of a scissor.
In simple words: Look at how the lines are placed. If they run next to each other and never touch, they are parallel. If they cross, they are intersecting. If they cross at a perfect square corner, they are perpendicular.

🎯 Exam Tip: To identify line types, imagine extending them indefinitely. If they never meet, they're parallel. If they cross, they intersect. If they form a 90-degree angle, they're perpendicular.

 

Question 2. Find the parallel and intersecting line segments in the picture given below.
Answer:
(a) Parallel line segments:
\( \overline{\mathrm{YZ}} \) and \( \overline{\mathrm{DE}} \)
\( \overline{\mathrm{EA}} \) and \( \overline{\mathrm{ZV}} \)
\( \overline{\mathrm{VW}} \) and \( \overline{\mathrm{AB}} \)
\( \overline{\mathrm{WX}} \) and \( \overline{\mathrm{BC}} \)
\( \overline{\mathrm{YX}} \) and \( \overline{\mathrm{DC}} \)
\( \overline{\mathrm{YD}} \) and \( \overline{\mathrm{XC}} \)
\( \overline{\mathrm{XC}} \) and \( \overline{\mathrm{WB}} \)
\( \overline{\mathrm{WB}} \) and \( \overline{\mathrm{VA}} \)
\( \overline{\mathrm{VA}} \) and \( \overline{\mathrm{ZE}} \)
\( \overline{\mathrm{ZE}} \) and \( \overline{\mathrm{YD}} \)
(b) Intersecting line segments:
\( \overline{\mathrm{DE}} \) and \( \overline{\mathrm{ZV}} \)
\( \overline{\mathrm{WX}} \) and \( \overline{\mathrm{DC}} \)
The parallel lines in this polygon form a symmetric pattern, making it a regular polygon when all sides and angles are equal.
In simple words: Parallel lines are like train tracks, always the same distance apart and never meeting. Intersecting lines are like crossroads, where two lines meet or cross over each other.

🎯 Exam Tip: When listing line segments, ensure you use the correct notation (a bar over the letters, like \( \overline{\mathrm{AB}} \)) and identify all possible pairs, not just a few.

 

Question 3. Name the following angles as shown in the figure.
(i) \( \angle 1 = \)
(ii) \( \angle 2 = \)
(iii) \( \angle 3 = \)
(iv) \( \angle 1 + \angle 2 = \)
(v) \( \angle 2 + \angle 3 = \)
(vi) \( \angle 1 + \angle 2 + \angle 3 = \)
Answer:
(i) \( \angle 1 = \angle DBC \) or \( \angle CBD \)
(ii) \( \angle 2 = \angle DBE \) or \( \angle EBD \)
(iii) \( \angle 3 = \angle ABE \) or \( \angle EBA \)
(iv) \( \angle 1 + \angle 2 = \angle EBC \) or \( \angle CBE \)
(v) \( \angle 2 + \angle 3 = \angle ABD \) or \( \angle DBA \)
(vi) \( \angle 1 + \angle 2 + \angle 3 = \angle ABC \) or \( \angle CBA \)
When naming angles, the middle letter always represents the vertex (the corner point) of the angle. The order of the other two letters doesn't change the angle itself.
In simple words: An angle is named by three letters. The middle letter is always where the two lines meet. When you add angles together, you get a bigger angle that covers the space of all the smaller ones.

🎯 Exam Tip: Always use three letters to name an angle clearly, with the vertex in the middle. For a single angle at a vertex, sometimes just the vertex letter is used if there's no confusion.

 

Question 4. Measure the angles of the given figures using a protractor and identify the type of angle as acute, obtuse, right or straight.
Answer:
(i) This is a right angle. It measures exactly \( 90^\circ \).
(ii) This is an acute angle. It measures less than \( 90^\circ \).
(iii) This is a straight angle. It measures exactly \( 180^\circ \).
(iv) This is an obtuse angle. It measures more than \( 90^\circ \) but less than \( 180^\circ \).
These angle types are fundamental in geometry and are used to describe shapes and their properties.
In simple words: We sort angles by how wide they are. A right angle is a perfect corner (\( 90^\circ \)). An acute angle is smaller than a corner. An obtuse angle is wider than a corner but not flat. A straight angle is a flat line (\( 180^\circ \)).

🎯 Exam Tip: Remember the key measures: Acute \( < 90^\circ \), Right \( = 90^\circ \), Obtuse \( > 90^\circ \) but \( < 180^\circ \), Straight \( = 180^\circ \). Using a protractor accurately is essential.

 

Question 5. Draw the following angles using the protractor.
(i) \( 45^\circ \)
(ii) \( 120^\circ \)
(iii) \( 65^\circ \)
(iv) \( 135^\circ \)
(v) \( 0^\circ \)
(vi) \( 180^\circ \)
(vii) \( 38^\circ \)
(viii) \( 90^\circ \)
Answer:
To draw these angles, first draw a ray (a line with one endpoint). Then, place the center of the protractor on the endpoint of the ray and align the ray with the \( 0^\circ \) mark. Finally, mark the degree indicated for each angle and draw a second ray from the endpoint through this mark. This creates the angle. For a \( 0^\circ \) angle, the two rays lie exactly on top of each other. For a \( 180^\circ \) angle, the two rays form a straight line pointing in opposite directions.
In simple words: To draw an angle, you use a special tool called a protractor. You make a starting line, then measure how wide the angle needs to be, and draw the second line.

🎯 Exam Tip: Make sure the protractor's center aligns perfectly with the vertex of your angle and the base line is exactly on the \( 0^\circ \) mark to get accurate measurements.

 

Question 6. From the figures given below, classify the following pairs of angles into non-complementary.
Answer:
We know that two angles are complementary if their sum is exactly \( 90^\circ \).
(a) The complementary pairs are:
(i) The two \( 45^\circ \) angles add up to \( 90^\circ \).
(v) The two \( 45^\circ \) angles in the right-angled figure also add up to \( 90^\circ \). These angles form a right angle together, making them complementary.
(b) The non-complementary pairs are:
(ii) The two \( 25^\circ \) angles add up to \( 50^\circ \), not \( 90^\circ \).
(iii) The \( 75^\circ \) and \( 105^\circ \) angles add up to \( 180^\circ \), not \( 90^\circ \).
(iv) The \( 40^\circ \) and \( 52^\circ \) angles add up to \( 92^\circ \), not \( 90^\circ \).
In simple words: Complementary angles are like two puzzle pieces that fit together to make a perfect corner (90 degrees). If they don't add up to 90 degrees, they are non-complementary.

🎯 Exam Tip: Always add the measures of the given angles. If the sum is exactly \( 90^\circ \), they are complementary. Otherwise, they are not.

 

Question 7. From the figures given below, classify the following pairs of angles into supplementary and non supplementary.
Answer:
We know that two angles are supplementary if their sum is exactly \( 180^\circ \).
The supplementary pairs are:
(ii) The \( 85^\circ \) and \( 95^\circ \) angles add up to \( 180^\circ \).
(iv) The \( 30^\circ \) and \( 150^\circ \) angles add up to \( 180^\circ \).
The non-supplementary pairs are:
(i) The \( 30^\circ \) and \( 120^\circ \) angles add up to \( 150^\circ \), not \( 180^\circ \).
(iii) The \( 120^\circ \) and \( 110^\circ \) angles add up to \( 230^\circ \), not \( 180^\circ \).
Supplementary angles often appear when two lines intersect, forming angles on a straight line.
In simple words: Supplementary angles are two angles that, when put together, make a straight line (180 degrees). If they don't add up to 180 degrees, they are not supplementary.

🎯 Exam Tip: For supplementary angles, the sum must be precisely \( 180^\circ \). Practice identifying these pairs quickly by summing the angle measures.

 

Question 8. From the figure (i) name a pair of complementary angles. (ii) name a pair of supplementary angles.
Answer:
(i) Pairs of complementary angles (angles that add up to \( 90^\circ \)):
\( \angle FAE \) and \( \angle EAD \)
\( \angle FAD \) and \( \angle DAC \)
(ii) Pairs of supplementary angles (angles that add up to \( 180^\circ \)):
\( \angle BAC \) and \( \angle CAE \)
\( \angle FAB \) and \( \angle BAC \)
\( \angle FAB \) and \( \angle FAE \)
In geometry, understanding how angles relate to each other in a figure is crucial for solving problems.
In simple words: Look for angles that add up to 90 degrees (complementary) or 180 degrees (supplementary). You can find these pairs by observing how the lines meet and form different angles.

🎯 Exam Tip: Identify all angles at a common vertex and then look for pairs whose sums are \( 90^\circ \) or \( 180^\circ \). Don't forget to consider adjacent angles.

 

Question 9. Find the complementary angle of
(i) \( 30^\circ \)
(ii) \( 26^\circ \)
(iii) \( 85^\circ \)
(iv) \( 0^\circ \)
(v) \( 90^\circ \)
Answer:

A complementary angle always adds up to \( 90^\circ \) with the given angle, completing a right angle.
In simple words: To find a complementary angle, just subtract the given angle from 90 degrees. The number you get is the complementary angle.

🎯 Exam Tip: The sum of complementary angles is always \( 90^\circ \). Double-check your subtraction to ensure accuracy.

 

Question 10. Find the supplementary angle of
(i) \( 70^\circ \)
(ii) \( 35^\circ \)
(iii) \( 165^\circ \)
(iv) \( 90^\circ \)
(v) \( 0^\circ \)
(vi) \( 180^\circ \)
(vii) \( 95^\circ \)
Answer:

A supplementary angle always adds up to \( 180^\circ \) with the given angle, forming a straight line.
In simple words: To find a supplementary angle, take the given angle and subtract it from 180 degrees. The result is the supplementary angle.

🎯 Exam Tip: The sum of supplementary angles is always \( 180^\circ \). This relationship is important for problems involving angles on a straight line.

Challenging Problems

 

Question 11. Think and write an object having
(i) Parallel lines (1) ........... (2) ........... (3) ...........
(ii) Perpendicular lines (1) ........... (2) ........... (3) ...........
(iii) Intersecting lines (1) ........... (2) ........... (3) ...........
Answer:
(i) Objects with parallel lines:
1. Legs of a table
2. Railway tracks
3. Edges of a ruler or scale
(ii) Objects with perpendicular lines:
1. Adjacent sides of a whiteboard or blackboard
2. Crossbars of windows
3. Adjacent sides of a textbook
(iii) Objects with intersecting lines:
1. Crossbars of windows (the lines themselves)
2. A ladder (the rungs and sides)
3. Blades of a scissor
Real-world examples help us understand geometric concepts in our daily lives.
In simple words: Parallel lines are like the sides of a road that never touch. Perpendicular lines are like the corners of a room, forming a perfect square angle. Intersecting lines are like roads crossing each other.

🎯 Exam Tip: Think of common household items or structures to easily recall examples for each type of line. Visualize the lines to ensure they fit the definition.

 

Question 12. Which angle is equal to twice its complement.
Answer:
Let the angle be \( x \).
The complement of the angle is \( (90^\circ - x) \).
According to the problem,
\( x = 2 \times (90^\circ - x) \)
\( x = 180^\circ - 2x \)
Now, bring all \( x \) terms to one side:
\( x + 2x = 180^\circ \)
\( 3x = 180^\circ \)
To find \( x \), divide both sides by 3:
\( x = \frac{180^\circ}{3} \)
\( x = 60^\circ \)
So, the angle is \( 60^\circ \). This means its complement is \( 30^\circ \), and \( 60^\circ \) is indeed twice \( 30^\circ \).
In simple words: If an angle is twice as big as its complementary angle (the angle that adds up to 90 degrees with it), then that angle must be 60 degrees.

🎯 Exam Tip: Clearly define your variables and set up the equation based on the word problem. Remember that complementary angles add up to \( 90^\circ \).

 

Question 13. Which angle is equal to two-thirds of its supplement.
Answer:
Let the angle be \( x \).
The supplement of the angle is \( (180^\circ - x) \).
According to the problem,
\( x = \frac{2}{3} \times (180^\circ - x) \)
First, multiply both sides by 3 to remove the fraction:
\( 3x = 2(180^\circ - x) \)
Distribute the 2 on the right side:
\( 3x = 360^\circ - 2x \)
Now, bring all \( x \) terms to one side:
\( 3x + 2x = 360^\circ \)
\( 5x = 360^\circ \)
To find \( x \), divide both sides by 5:
\( x = \frac{360^\circ}{5} \)
\( x = 72^\circ \)
So, the angle is \( 72^\circ \). This angle is two-thirds of its supplementary angle, which is \( 108^\circ \).
In simple words: If an angle is two-thirds the size of its supplementary angle (the angle that adds up to 180 degrees with it), then that angle is 72 degrees.

🎯 Exam Tip: Remember that supplementary angles add up to \( 180^\circ \). Pay close attention to fraction multiplication and distributing terms when solving such equations.

 

Question 14. Given two angles are supplementary and one angle is \( 20^\circ \) more than the other. Find the two angles.
Answer:
Let the smaller angle be \( x \).
Then the other angle is \( x + 20^\circ \).
Since the two angles are supplementary, their sum is \( 180^\circ \).
According to the problem,
\( x + (x + 20^\circ) = 180^\circ \)
Combine the \( x \) terms:
\( 2x + 20^\circ = 180^\circ \)
Subtract \( 20^\circ \) from both sides:
\( 2x = 180^\circ - 20^\circ \)
\( 2x = 160^\circ \)
To find \( x \), divide both sides by 2:
\( x = \frac{160^\circ}{2} \)
\( x = 80^\circ \)
Now find the second angle:
\( x + 20^\circ = 80^\circ + 20^\circ \)
\( = 100^\circ \)
Therefore, the two angles are \( 80^\circ \) and \( 100^\circ \). These angles correctly add up to \( 180^\circ \).
In simple words: When two angles add up to 180 degrees, and one is 20 degrees bigger than the other, the two angles are 80 degrees and 100 degrees.

🎯 Exam Tip: When dealing with two unknown values, assign a variable to one and express the other in terms of that variable based on the problem's conditions. Then set up the equation using the definition of supplementary angles.

 

Question 15. Two complementary angles are in ratio 7 : 2. Find the angles.
Answer:
Let the two complementary angles be \( 7x \) and \( 2x \).
Since they are complementary, their sum is \( 90^\circ \).
According to the problem,
\( 7x + 2x = 90^\circ \)
Combine the \( x \) terms:
\( 9x = 90^\circ \)
To find \( x \), divide both sides by 9:
\( x = \frac{90^\circ}{9} \)
\( x = 10^\circ \)
Now, find the measure of each angle:
First angle: \( 7x = 7 \times 10^\circ = 70^\circ \)
Second angle: \( 2x = 2 \times 10^\circ = 20^\circ \)
Therefore, the two angles are \( 70^\circ \) and \( 20^\circ \). These angles add up to \( 90^\circ \), confirming they are complementary.
In simple words: If two complementary angles are in a 7:2 ratio, it means one angle is 70 degrees and the other is 20 degrees.

🎯 Exam Tip: When angles are given in a ratio, represent them as multiples of a variable (e.g., \( 7x \) and \( 2x \)). Then use the definition of complementary or supplementary angles to form an equation.

 

Question 16. Two supplementary angles are in ratio 5 : 4. Find the angles.
Answer:
Let the two supplementary angles be \( 5x \) and \( 4x \).
Since they are supplementary, their total sum is \( 180^\circ \).
Given that the angles are in the ratio 5:4, the total parts are \( 5 + 4 = 9 \) equal parts.
So, \( 5x + 4x = 180^\circ \)
\( 9x = 180^\circ \)
To find \( x \), divide both sides by 9:
\( x = \frac{180^\circ}{9} \)
\( x = 20^\circ \)
Now, find the measure of each angle:
One angle: \( 5x = 5 \times 20^\circ = 100^\circ \)
Another angle: \( 4x = 4 \times 20^\circ = 80^\circ \)
The two angles are \( 100^\circ \) and \( 80^\circ \). Their sum is \( 100^\circ + 80^\circ = 180^\circ \), which is correct for supplementary angles.
In simple words: If two supplementary angles are in a 5:4 ratio, it means one angle is 100 degrees and the other is 80 degrees.

🎯 Exam Tip: Always represent angles in a ratio as variable multiples (e.g., \( 5x, 4x \)). Ensure the sum matches the definition ( \( 90^\circ \) for complementary, \( 180^\circ \) for supplementary) before solving for the variable.