Samacheer Kalvi Class 6 Maths Solutions Term 1 Chapter 4 Geometry Exercise 4.2

Question 1. Use any number of the given dots to make different angles.
(i) An Acute angle
(ii) An Obtuse Angle
(iii) A Right Angle
(iv) A Straight Angle
Answer: To form different types of angles using dots, we can pick three dots to define the vertex and two rays.
(i) An Acute Angle: An acute angle is smaller than a right angle (less than \( 90^\circ \)).
(ii) An Obtuse Angle: An obtuse angle is larger than a right angle but smaller than a straight angle (between \( 90^\circ \) and \( 180^\circ \)).
(iii) A Right Angle: A right angle is exactly \( 90^\circ \).
(iv) A Straight Angle: A straight angle is exactly \( 180^\circ \), forming a straight line.In simple words: To make angles, pick three dots: one for the corner (vertex) and two others to show the lines (rays). An acute angle is a small opening, an obtuse angle is a wide opening, a right angle is a perfect square corner, and a straight angle is a flat line.

🎯 Exam Tip: When drawing angles, clearly mark the vertex and the rays. Use an arc to indicate the angle, and a square symbol for a right angle.

Question 2. Name the vertex and sides that form each angle.
Answer: We identify the vertex as the common endpoint of the two rays, and the sides as the rays themselves.
(i) The vertex is \( D \), and the sides are \( DE \) and \( DF \).
(ii) The vertex is \( D \), and the sides are \( DE \) and \( DC \).
(iii) The vertex is \( P \), and the sides are \( PQ \) and \( PR \).
(iv) The vertex is \( S \), and the sides are \( SV \) and \( ST \).In simple words: The vertex is the corner point where the two lines of an angle meet. The sides of the angle are the two lines (or rays) that start from that corner point.

🎯 Exam Tip: Remember that the vertex is always the middle letter when naming an angle with three letters (e.g., \( \angle EDF \)).

Question 3. Pick out the Right angles from the given figures.
Answer: The right angles from the given figures are:
(i) \( 90^\circ \)
(iii) The angle marked with a square symbol, indicating \( 90^\circ \).
(v) The angle marked with a square symbol, indicating \( 90^\circ \). Right angles are exactly \( 90^\circ \) and are often shown with a small square at the vertex.In simple words: We look for angles that are exactly 90 degrees. These are usually shown with a small square mark at their corner.

🎯 Exam Tip: A square symbol at the vertex is a universal sign for a right angle, meaning it measures exactly 90 degrees.

Question 4. Pick out the Acute angles from the given figures.
Answer: An acute angle is an angle that measures less than \( 90^\circ \).
The acute angles from the given figures are:
(i) \( 33^\circ \)
(iii) \( 56^\circ \)
(iv) \( 89^\circ \) These angles are all smaller than \( 90^\circ \).In simple words: Acute angles are the "sharp" angles, meaning they are smaller than a right angle (less than 90 degrees).

🎯 Exam Tip: To identify acute angles, simply check if their measure is less than 90 degrees. They look smaller than a perfect corner.

Question 5. Pick out the Obtuse angles from the given figures.
Answer: An obtuse angle is an angle that measures more than \( 90^\circ \) but less than \( 180^\circ \).
The obtuse angles from the given figures are:
(i) \( 101^\circ \)
(ii) \( 129^\circ \) These angles are all larger than \( 90^\circ \) but less than a straight line.In simple words: Obtuse angles are the "wide" angles. They are bigger than a right angle (more than 90 degrees) but not as big as a straight line (less than 180 degrees).

🎯 Exam Tip: Visually, an obtuse angle appears wider than an L-shape (right angle) but not completely flat (straight angle).

Question 6. Name the angle in each figure given below in all the possible ways.
Answer: Angles can be named in three ways: by its vertex, by a number inside it, or by three letters with the vertex in the middle.
(i) \( \angle M \) or \( \angle LMN \) or \( \angle NML \)
(ii) \( \angle Q \) or \( \angle PQR \) or \( \angle RQP \)
(iii) \( \angle N \) or \( \angle MNO \) or \( \angle ONM \)
(iv) \( \angle A \) or \( \angle TAS \) or \( \angle SAT \)
(v) \( \angle Y \) or \( \angle XYZ \) or \( \angle ZYX \)
(vi) There are 3 angles in this figure:
• \( \angle ADC \) or \( \angle CDA \)
• \( \angle CDB \) or \( \angle BDC \)
• \( \angle ADB \) or \( \angle BDA \)In simple words: You can name an angle by just its corner letter, or by using three letters where the middle letter is always the corner. If there are many angles at one corner, use three letters so everyone knows which angle you mean.

🎯 Exam Tip: When using three letters to name an angle, always place the vertex letter in the middle. This clearly indicates the angle being referred to.

Question 7. Say True or False.
(i) \( 20^\circ \) and \( 70^\circ \) are complementary.
(ii) \( 88^\circ \) and \( 12^\circ \) are complementary.
(iii) \( 80^\circ \) and \( 180^\circ \) are supplementary.
(iv) \( 0^\circ \) and \( 180^\circ \) are supplementary.
Answer:
(i) True. Complementary angles add up to \( 90^\circ \). Here, \( 20^\circ + 70^\circ = 90^\circ \).
(ii) False. Complementary angles must sum to \( 90^\circ \). Here, \( 88^\circ + 12^\circ = 100^\circ \), which is not \( 90^\circ \).
(iii) False. Supplementary angles add up to \( 180^\circ \). Here, \( 80^\circ + 180^\circ = 260^\circ \), which is not \( 180^\circ \).
(iv) True. Supplementary angles add up to \( 180^\circ \). Here, \( 0^\circ + 180^\circ = 180^\circ \). A quick way to remember is 'C' for complementary comes before 'S' for supplementary, just as 90 comes before 180.In simple words: Two angles are complementary if they add up to 90 degrees. Two angles are supplementary if they add up to 180 degrees. You just need to check if their sum matches these numbers.

🎯 Exam Tip: Always remember that complementary angles sum to 90 degrees, and supplementary angles sum to 180 degrees. Use these definitions to verify the statements.

Question 8. Draw and label each of the angles.
(i) \( \angle NAS = 90^\circ \)
(ii) \( \angle BIG = 35^\circ \)
(iii) \( \angle SMC = 145^\circ \)
Answer: Drawing angles accurately helps in understanding their properties.
(i) \( \angle NAS = 90^\circ \) (A Right Angle)
(ii) \( \angle BIG = 35^\circ \) (An Acute Angle)
(iii) \( \angle SMC = 145^\circ \) (An Obtuse Angle)In simple words: To draw angles, you need a protractor. Place the center of the protractor on the vertex, align one line with zero, and then mark the degree for the other line.

🎯 Exam Tip: Use a protractor to draw angles accurately. For a right angle, you can also use the corner of a square object or a set square.

Question 9. Identify the types of angles shown by the hands of the given clock.
Answer: The hands of a clock form different angles based on their positions.
(i) The hands show an Obtuse angle (e.g., between 9 and 2).
(ii) The hands show a Zero angle (both hands point to the same number, e.g., 12).
(iii) The hands show a Straight angle (e.g., 6 o'clock or 12:30).
(iv) The hands show an Acute angle (e.g., between 12 and 1).
(v) The hands show a Right angle (e.g., 3 o'clock or 9 o'clock).In simple words: Look at how wide open the clock hands are. A small opening is acute, a very wide opening is obtuse, a straight line is a straight angle, and when hands are on top of each other, it's a zero angle. A perfect L-shape is a right angle.

🎯 Exam Tip: Visualize the angle formed by the clock hands. Remember that each hour mark represents \( 30^\circ \) ( \( 360^\circ / 12 = 30^\circ \)).

Question 10. Find the supplementary/complementary angles in each case.
Answer: We use the definitions of complementary (sum to \( 90^\circ \)) and supplementary (sum to \( 180^\circ \)) angles to find the missing values.
(i) Since \( \angle ABD \) and \( \angle DBC \) form a straight line, they are supplementary angles.
\( \angle ABD + \angle DBC = 180^\circ \)
\( \angle ABD + 25^\circ = 180^\circ \)
\( \implies \angle ABD = 180^\circ - 25^\circ \)
\( \implies \angle ABD = 155^\circ \)
(ii) Since \( \angle ABD \) and \( \angle DBC \) form a right angle, they are complementary angles.
\( \angle ABD + \angle DBC = 90^\circ \)
\( \angle ABD + 30^\circ = 90^\circ \)
\( \implies \angle ABD = 90^\circ - 30^\circ \)
\( \implies \angle ABD = 60^\circ \)
(iii) Since \( \angle ABD \) and \( \angle DBC \) form a right angle, they are complementary angles.
\( \angle ABD + \angle DBC = 90^\circ \)
\( \angle ABD + 46^\circ = 90^\circ \)
\( \implies \angle ABD = 90^\circ - 46^\circ \)
\( \implies \angle ABD = 44^\circ \)
(iv) Since \( \angle CBD \) and \( \angle EBC \) form a straight line, they are supplementary angles.
\( \angle CBD + \angle EBC = 180^\circ \)
\( 67^\circ + \angle EBC = 180^\circ \)
\( \implies \angle EBC = 180^\circ - 67^\circ \)
\( \implies \angle EBC = 113^\circ \)In simple words: If two angles together make a straight line (180 degrees), they are supplementary. If they together make a perfect corner (90 degrees), they are complementary. To find a missing angle, subtract the known angle from 90 or 180 degrees, depending on the case.

🎯 Exam Tip: Always identify if the angles are complementary (sum to \( 90^\circ \)) or supplementary (sum to \( 180^\circ \)) first, based on the diagram or given information, before performing calculations.

Question 11. In this figure, which is not the correct way of naming an angle?
(a) \( \angle Y \)
(b) \( \angle ZXY \)
(c) \( \angle ZYX \)
(d) \( \angle XYZ \)
Answer: (b) \( \angle ZXY \)
In simple words: When you name an angle with three letters, the middle letter must always be the corner point (vertex) of the angle. Here, 'Y' is the vertex, so 'X' should not be in the middle.

🎯 Exam Tip: The middle letter in a three-letter angle name always represents the vertex (the point where the two rays meet). If the vertex is not the middle letter, the naming is incorrect.

Question 12. In this figure, \( \angle AYZ = 45^\circ \). If point 'A' is shifted to point 'B' along the ray, then the measure of \( \angle BYZ \) is
(a) more than \( 45^\circ \)
(b) \( 45^\circ \)
(c) less than \( 45^\circ \)
(d) \( 90^\circ \)
Answer: (b) \( 45^\circ \)
In simple words: Moving a point along one of the angle's lines (rays) does not change the angle's measurement. The angle stays the same because the lines are still opening by the same amount from the corner.

🎯 Exam Tip: The measure of an angle depends only on the separation between its two rays at the vertex, not on the length of the rays or the position of points along them.