Maharashtra Board Class 7 Chapter 4 Set 20 Angles and Pairs of Angles Solutions

Question 1. Lines AC and BD intersect at point P. m∠APD = 47° Find the measures of ∠APB, ∠BPC, ∠CPD.
ℹ️ चित्र व्याख्या (Diagram Explanation): दो सीधी रेखाएँ AC और BD बिंदु P पर प्रतिच्छेद कर रही हैं। कोण APD का माप 47 डिग्री दिखाया गया है। यह चित्र रैखिक युग्म और ऊर्ध्वाधर विपरीत कोणों को समझने में मदद करता है।
Answer: ∠APD and ∠APB are angles in a linear pair.
\( m∠APD + m∠APB = 180° \)
∴ \( 47 + m∠APB = 180 \)
∴ \( 47 + m∠APB - 47 = 180 - 47 \) ....(Subtracting 47 from both sides)
∴ \( m∠APB = 133° \)
\( m∠CPD = m∠APB = 133° \) ... .(Vertically opposite angles)
\( m∠BPC = m∠APD = 47° \) .(Vertically opposite angles)
The measures of ∠APB, ∠BPC and ∠CPD are 133°, 47° and 133° respectively.
In simple words: When two lines intersect, angles forming a linear pair add up to 180°, and vertically opposite angles are equal. Using the given angle, we find the other angles by applying these geometric properties.

🎯 Exam Tip: Remember to correctly identify linear pairs and vertically opposite angles to calculate unknown angle measures. Clearly state the reasons for each step in your solution.

Question 2. Lines PQ and RS intersect at point M. m∠PMR = x°.What are the measures of ∠PMS, ∠SMQ and ∠QMR?
ℹ️ चित्र व्याख्या (Diagram Explanation): दो सीधी रेखाएँ PQ और RS बिंदु M पर प्रतिच्छेद कर रही हैं। कोण PMR का माप x डिग्री दर्शाया गया है। यह चित्र कोणों के रैखिक युग्म और ऊर्ध्वाधर विपरीत कोणों के संबंधों को दर्शाता है।
Answer: ∠PMR and ∠PMS are angles in a linear pair.
∴ \( m∠PMR + m∠PMS = 180° \)
∴ \( x + m∠PMS = 180 \)
∴ \( m∠PMS = (180-x)° \)
\( m∠QMR = m∠PMS = (180 - x)° \) ... .(Vertically opposite angles)
\( m∠SMQ = m∠PMR = x° \) (Vertically opposite angles)
The measures of ∠PMS, ∠SMQ and ∠QMR are (180 - x)°, x° and (180 - x)° respectively.
In simple words: When two lines intersect, if one angle is 'x', its linear pair will be (180-x), and its vertically opposite angle will also be 'x'. We use these rules to find all unknown angles in terms of 'x'.

🎯 Exam Tip: Clearly state the relationship between angles (linear pair, vertically opposite) when deriving their measures. Express the unknown angles in terms of the given variable 'x'.