Maharashtra Board Class 7 Chapter 4 Set 17 Angles and Pairs of Angles Solutions

Question 1. Write the measures of the supplements of the angles given below:
(i) 15°
(ii) 85°
(iii) 120°
(iv) 37°
(v) 108°
(vi) 0°
(vii) a°
Answer:
(i) Let the measure of the supplementary angle be x°.
.. \(15 + x = 180\)
.. \(15 + x - 15 = 180 - 15\)
....(Subtracting 15 from both sides)
.. \(x = 165\)
.. The measures of the supplement of an angle of 15° is 165°.
(ii) Let the measure of the supplementary angle be x°.
.. \(85 + x = 180\)
.. \(85 + x - 85 = 180 - 85\)
....(Subtracting 85 from both sides)
.. \(x = 95\)
.. The measures of the supplement of an angle of 85° is 95°.
(iii) Let the measure of the supplementary angle be x°.
.. \(120 + x = 180\)
.. \(120 + x - 120 = 180 - 120\)
....(Subtracting 120 from both sides)
.. \(x = 60\)
.. The measures of the supplement of an angle of 120° is 60°.
(iv) Let the measure of the supplementary angle be x°.
.. \(37 + x = 180\)
.. \(37 + x - 37 = 180 - 37\)
....(Subtracting 37 from both sides)
.. \(x = 143\)
.. The measures of the supplement of an angle of 37° is 143°.
(v) Let the measure of the supplementary angle be x°.
.. \(108 + x = 180\)
.. \(108 + x - 108 = 180 - 108\)
....(Subtracting 108 from both sides)
.. \(x = 72\)
.. The measures of the supplement of an angle of 108° is 72°.
(vi) Let the measure of the supplementary angle be x°.
.. \(0 + x = 180\)
.. \(x = 180\)
.. The measures of the supplement of an angle of 0° is 180°.
(vii) Let the measure of the supplementary angle be x°.
.. \(a + x = 180\)
∴\(a + x - a = 180 - a\)
....(Subtracting a from both sides) \(x = (180 - a)\)
.. The measures of the supplement of an angle of a° is \((180 - a)°\).
In simple words: To find the supplement of an angle, subtract its measure from 180 degrees. This is because supplementary angles always add up to 180 degrees.

🎯 Exam Tip: Remember the definition of supplementary angles (sum is 180°) and complementary angles (sum is 90°) to quickly solve these types of problems.

Question 2. The measures of some angles are given below. Use them to make pairs of complementary and supplementary angles.
m∠B = 60°
m∠N = 30°
m∠Y = 90°
m∠J = 150°
m∠D = 75°
m∠E = 0°
m∠F = 15°
m∠G = 120°
Answer:
(i) \(m∠B + m∠N = 60° + 30°\)
\( = 90°\)
∴∠B and ∠N are a pair of complementary angles.
(ii) \(m∠Y + m∠E = 90° + 0°\)
\( = 90°\)
∴∠Y and ∠E are a pair of complementary angles.
(iii) \(m∠D + m∠F = 75° + 15°\)
\( = 90°\)
∴∠D and ∠F are a pair of complementary angles.
(iv) \(m∠B + m∠G = 60° + 120°\)
\( = 180°\)
∴∠B and ∠G are a pair of supplementary angles.
(v) \(m∠N + m∠J = 30° + 150°\)
\( = 180°\)
∴∠N and ∠J are a pair of supplementary angles.
In simple words: Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees. We identify pairs from the given list that satisfy these sums.

🎯 Exam Tip: Clearly distinguish between complementary (sum 90°) and supplementary (sum 180°) angles. Show your addition steps for full credit.

Question 3. In ΔXYZ, m∠Y = 90°. What kind of a pair do ∠X and ∠Z make?
Answer:
In ΔΧΥΖ,
\(m∠X + m∠Y + m∠Z = 180°\) ....(Sum of the measure of the angles of a triangle is 180°)
\(m∠X + 90 + m∠Z = 180\)
∴\(m∠X + 90 + m∠Z - 90 = 180 - 90\) ....(Subtracting 90 from both sides)
∴\(m∠X + m∠Z = 90°\)
∴∠X and ∠Z make a pair of complementary angles.
In simple words: Since the sum of angles in a triangle is 180 degrees and one angle is 90 degrees, the other two angles must add up to 90 degrees, making them complementary.

🎯 Exam Tip: Always state the geometric properties you are using (e.g., sum of angles in a triangle) to justify your steps.

Question 4. The difference between the measures of the two angles of a complementary pair is 40°. Find the measures of the two angles.
Answer:
Let the measure of one angle be x°.
Measure of other angle = \((x + 40)°\)
\(x + (x + 40) = 90\) ...(Since, the two angles are complementary)
.. \(2x + 40 - 40 = 90 - 40\) ....(Subtracting 40 from both sides)
.. \(2x = 50\)
.. \(x = \frac{50}{2}\)
.. \(x = 25\)
.. \(x + 40 = 25 + 40\)
\( = 65\)
The measures of the two angles is 25° and 65°.
In simple words: We set up an equation where one angle is 'x' and the other is 'x + 40', knowing they sum to 90 degrees. Solving this equation gives us the values of both angles.

🎯 Exam Tip: Define your variables clearly. Ensure your final answers address all parts of the question, and double-check your arithmetic.

Question 5. *PTNM is a rectangle. Write the names of the pairs of supplementary angles.

ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में PTNM एक आयत दर्शाया गया है। आयत के चारों कोने P, T, N, और M हैं। यह एक बंद चतुर्भुज है जिसमें सभी कोण 90 डिग्री के होते हैं।
Answer:
Since, each angle of the rectangle is 90°.
.. Pairs of supplementary angles are:
(i) ∠P and ∠M
(ii) ∠P and ∠N
(iii) ∠P and ∠T
(iv) ∠M and ∠N
(v) ∠M and ∠T
(vi) ∠N and ∠T
In simple words: In a rectangle, all angles are 90 degrees. Any two angles that add up to 180 degrees are supplementary, and in a rectangle, any two adjacent or opposite angles are 90° + 90° = 180°, making them supplementary.

🎯 Exam Tip: Recall the properties of rectangles: all interior angles are 90°. Any two angles will be supplementary since 90° + 90° = 180°.

Question 6. If m∠A = 70°, what is the measure of the supplement of the complement of ∠A?
Answer:
Let the measure of the complement of ∠A be x° and the measure of its supplementary angle be y°.
\(m∠A + x = 90°\)
.. \(70 + x = 90\)
.. \(70 + x - 70 = 90 - 70\) ....(Subtracting 70 from both sides)
.. \(x = 20\)
Since, x and y are supplementary angles.
.. \(x + y = 180\)
.. \(20 + y = 180\)
.. \(20 + y - 20 = 180 - 20\) ....(Subtracting 20 from both sides)
.. \(y = 160\)
.. The measure of supplement of the complement of ∠A is 160°.
In simple words: First, find the complement of ∠A by subtracting 70° from 90°. Then, find the supplement of that complementary angle by subtracting it from 180°.

🎯 Exam Tip: Break down complex problems into smaller, manageable steps. First find the complement, then its supplement. Clearly label each intermediate value.

Question 7. If ∠A and ∠B are supplementary angles and m∠B = (x + 20)°, then what would be m∠A?
Answer:
Since, ∠A and ∠B are supplementary angles.
∴\(m∠A + m∠B = 180\)
\(m∠A + x + 20 = 180\)
∴\(m∠A + x + 20 - 20 = 180 - 20\) ....(Subtracting 20 from both sides)
∴\(m∠A + x = 160\)
\(m∠A + x - x = 160 - x\) ....(Subtracting x from both sides)
∴\(m∠A = (160 - x)°\)
The measure of ∠A is \((160 - x)°\).
In simple words: Since ∠A and ∠B are supplementary, their sum is 180 degrees. Substitute the given expression for m∠B and solve for m∠A by isolating it in the equation.

🎯 Exam Tip: When dealing with algebraic expressions for angles, treat them like any other variable. Set up the equation based on the definition of supplementary angles and solve for the unknown.

Maharashtra Board Class 7 Maths Chapter 4 Angles And Pairs Of Angles Practice Set 17 Intext Questions And Activities

Question 1. Observe the figure and answer the following questions. T is a point on line AB.

ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में एक सीधी रेखा AB है, जिस पर एक बिंदु T स्थित है। बिंदु T से एक किरण निकल रही है जो रेखा AB के ऊपर स्थित है, जिससे एक कोण ∠ATB बनता दिख रहा है। यह कोण एक सीधी रेखा पर बनता है।
1. What kind of angle is ∠ATB?
2. What is its measure?
Answer:
1. Straight angle
2. 180°
In simple words: An angle formed by a straight line is called a straight angle, and its measure is always 180 degrees.

🎯 Exam Tip: Recognize common angles like straight angles (180°) and right angles (90°). This knowledge helps in understanding basic geometric diagrams.