Maharashtra Board Class 7 Chapter 4 Set 21 Angles and Pairs of Angles Solutions

Question 1. ∠ACD is an exterior angle of ∆ABC. The measures of ∠A and ∠B are equal. If m∠ACD = 140°, find the measures of the angles ∠A and ∠B.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक त्रिभुज ABC है। बिंदु C से एक रेखा D तक विस्तारित है, जिससे ∠ACD त्रिभुज का बाह्य कोण बनता है। ∠A और ∠B त्रिभुज के आंतरिक कोण हैं। कोण B पर, रेखा BC है, और कोण C पर, रेखा AC है जो D तक विस्तारित है।
Answer: Let the measures of ∠A be x°. \(m\angle A = m\angle B = x^\circ\) ∠ACD is the exterior angle of ∆ABC
Therefore, \(m\angle ACD = m\angle A + m\angle B\)
\(140 = x + x\)
\(140 = 2x\)
\(2x = 140\)
\(x = \frac{140}{2}\)
\( = 70\)
Therefore, The measures of the angles ∠A and ∠B is 70° each.
In simple words: The exterior angle of a triangle is equal to the sum of its two remote interior angles. Since the two interior angles (∠A and ∠B) are equal and their sum is 140°, each angle must be 70°.

🎯 Exam Tip: Remember the exterior angle theorem to quickly solve such problems. Clearly state the given conditions and the theorem used for full marks.

Question 2. Using the measures of the angles given in the figure alongside, find the measures of the remaining three angles.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक बिंदु O पर प्रतिच्छेद करती हुई दो सीधी रेखाओं AF और DE को दर्शाता है, जिससे चार कोण बनते हैं। कोण FOE का माप 4y है, कोण EOD का माप 8y है, और कोण DOC का माप 6y है। यह व्यवस्था शीर्षाभिमुख कोणों और रैखिक युग्मों को समझने में मदद करती है।
Answer: \(m\angle EOD = m\angle AOB = 8y\) ....(vertically opposite angles) ∠FOL, ∠EOD and ∠COD form a straight angle.
Therefore, \(m\angle FOE + m\angle EOD + m\angle COD = 180^\circ\)
\(4y + 8y + 6y = 180\)
\(18y = 180\)
Therefore, \(y = \frac{180}{18}\)
Therefore, \(y = 10\) \(m\angle EOD = 8y = 8 \times 10 = 80^\circ\) \(m\angle AOF = m\angle COD\) ....(Vertically opposite angles) \( = 6y = 6 \times 10 = 60^\circ\) \(m\angle BOC = m\angle FOE\) ....(Vertically opposite angles) \( = 4y = 4 \times 10 = 40^\circ\)
Therefore, The measures of ∠EOD, ∠AOF and ∠BOC are 80°, 60° and 40° respectively.
In simple words: By using the properties of vertically opposite angles and angles on a straight line, we can first find the value of 'y' and then calculate the measure of each unknown angle. Vertically opposite angles are equal, and angles forming a linear pair sum to 180°.

🎯 Exam Tip: Clearly identify and use the correct angle relationships (vertically opposite, angles on a straight line) to set up equations and solve for variables. Show all steps for clarity.

Question 3. In the isosceles triangle ABC, ∠A and ∠B are equal. ∠ACD is an exterior angle of ∆ABC. The measures of ∠ACB and ∠ACD are \((3x - 17)^\circ\) and \((8x + 10)^\circ\) respectively. Find the measures of ∠ACB and ∠ACD. Also find the measures of ∠A and ∠B.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक समद्विबाहु त्रिभुज ABC है, जहाँ कोण A और B समान हैं। कोण C पर, एक रेखा D तक विस्तारित है, जिससे ∠ACD बाह्य कोण बनता है। आंतरिक कोण ACB और बाह्य कोण ACD के माप बीजीय व्यंजकों \((3x - 17)^\circ\) और \((8x + 10)^\circ\) के रूप में दिए गए हैं।
Answer: Let the measure of ∠A be y°. Therefore, \(m\angle A = m\angle B = y^\circ\) ∠ACB and ∠ACD form a pair of linear angles.
Therefore, \(m\angle ACB + m\angle ACD = 180^\circ\)
\((3x - 17) + (8x + 10) = 180\)
\(3x + 8x - 17 + 10 = 180\)
\(11x - 7 = 180\)
\(11x - 7 + 7 = 180 + 7\) ...(Adding 7 on both sides.)
\(11x = 187\)
Therefore, \(x = \frac{187}{11} = 17\) \(m\angle ACB = 3x - 17 = (3 \times 17) - 17 = 51 - 17 = 34^\circ\) \(m\angle ACD = 8x + 10 = 8 \times 17 + 10 = 136 + 10 = 146^\circ\) Here ∠ACD is the exterior angle of ∆ABC and ∠A and ∠B are its remote interior angles.
Therefore, \(m\angle ACD = m\angle A + m\angle B\)
\(146 = y + y\)
\(146 = 2y\)
\(2y = 146\)
Therefore, \(y = \frac{146}{2} = 73\)
Therefore, The measures of ∠ACB, ∠ACD, ∠A and ∠B are 34°, 146°, 73° and 73° respectively.
In simple words: First, we use the property of linear pairs (∠ACB and ∠ACD sum to 180°) to find the value of x. Then, we calculate the measures of ∠ACB and ∠ACD. Finally, using the exterior angle theorem (∠ACD equals the sum of remote interior angles ∠A and ∠B), and knowing ∠A = ∠B, we find the measures of ∠A and ∠B.

🎯 Exam Tip: This problem combines linear pair properties with the exterior angle theorem. Ensure you correctly substitute the value of 'x' to find the angles and clearly state the theorems applied.

Maharashtra Board Class 7 Maths Chapter 4 Angles And Pairs Of Angles Practice Set 21 Intext Questions And Activities

Question 1. Use straws or sticks to make all the kinds of angles that you have learnt about. (Textbook pg. no. 29)
Answer: (Student should attempt the activity on their own)
In simple words: Students should use physical objects like straws or sticks to model and visualize different types of angles they have studied, such as acute, obtuse, right, straight, and reflex angles.

🎯 Exam Tip: Practical activities like this help in understanding geometric concepts better. While not directly tested, the experience reinforces theoretical knowledge.

Question 2. Observe the table given below and draw your conclusions (Textbook pg. no. 31)
Answer:

Solution:
(i) \(180^\circ\)
(ii) \(360^\circ\)
(iii) \(540^\circ\)
(iv) \(720^\circ\)
(v) \(180^\circ \times 5 = 900^\circ\)
(vi)
ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक नियमित षट्भुज को दर्शाता है, जिसमें छह समान भुजाएँ और छह समान कोण होते हैं, जिसे छह त्रिभुजों में विभाजित किया जा सकता है, जिनके केंद्र में एक सामान्य बिंदु है, या एक शीर्ष से विकर्णों द्वारा चार त्रिभुजों में। \(180^\circ \times 6 = 1080^\circ\)
In simple words: This table demonstrates that any polygon with 'n' sides can be divided into \((n-2)\) triangles by drawing diagonals from one vertex. Since the sum of angles in each triangle is 180°, the sum of interior angles of a polygon with 'n' sides is given by the formula \((n-2) \times 180^\circ\).

🎯 Exam Tip: Understand the formula for the sum of interior angles of a polygon: \((n-2) \times 180^\circ\), where 'n' is the number of sides. This formula is crucial for solving problems related to polygon angles.