Let's solve each problem step by step using the concepts of
linear pairs,
adjacent angles, and
vertically opposite angles.
---
🔹 1. First Figure (Top Left)
- Angles: 60° and $ x^\circ $ form a linear pair (they are adjacent and on a straight line).
- So, their sum is $ 180^\circ $.
$$
60 + x = 180 \Rightarrow x = 120
$$
✔ Answer: $ x = 120 $
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🔹 2. Second Figure (Top Middle)
- Angles: $ 3x^\circ $ and $ 2x^\circ $ form a linear pair.
$$
3x + 2x = 180 \Rightarrow 5x = 180 \Rightarrow x = 36
$$
✔ Answer: $ x = 36 $
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🔹 3. Third Figure (Top Right)
- Angles: $ 35^\circ $, $ x^\circ $, and $ 60^\circ $ are on a straight line.
- Sum of angles on a straight line = $ 180^\circ $
$$
35 + x + 60 = 180 \Rightarrow x = 180 - 95 = 85
$$
✔ Answer: $ x = 85 $
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🔹 4. Fourth Figure (Middle Left)
- Angles around point O: $ 3x^\circ $, $ 2x^\circ $, $ x^\circ $, and $ 2x^\circ $ form a full circle (360°).
$$
3x + 2x + x + 2x = 360 \Rightarrow 8x = 360 \Rightarrow x = 45
$$
✔ Answer: $ x = 45 $
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🔹 5. Fifth Figure (Middle Center)
- Vertically opposite angles: $ 105^\circ $ and $ 3x^\circ $ are vertically opposite → equal.
$$
3x = 105 \Rightarrow x = 35
$$
✔ Answer: $ x = 35 $
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🔹 6. Sixth Figure (Middle Right)
- Given: $ x = 25 $, find $ y $
- Angles: $ (2y + 5)^\circ $ and $ 3x^\circ $ are on a straight line → linear pair.
- Since $ x = 25 $, then $ 3x = 75^\circ $
$$
(2y + 5) + 75 = 180 \Rightarrow 2y + 80 = 180 \Rightarrow 2y = 100 \Rightarrow y = 50
$$
✔ Answer: $ x = 25 $, $ y = 50 $
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🔹 7. Seventh Figure (Bottom Left)
- Three angles at point O: $ (x+10)^\circ $, $ x^\circ $, $ (x+20)^\circ $
- They form a straight line → sum = $ 180^\circ $
$$
(x+10) + x + (x+20) = 180 \Rightarrow 3x + 30 = 180 \Rightarrow 3x = 150 \Rightarrow x = 50
$$
✔ Answer: $ x = 50 $
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🔹 8. Eighth Figure (Bottom Center)
- Angles around point O:
- $ 3x^\circ $, $ 3x^\circ $, and $ x^\circ $ — these three make a full circle?
- But notice: two $ 3x $ angles are vertically opposite and one $ x $ angle is below.
- Actually, from the diagram: the vertical angles are equal, and the angles on a straight line sum to 180°.
- Let’s analyze: The horizontal line $ C-A $ has two angles: $ 3x $ and $ x $. They form a linear pair.
$$
3x + x = 180 \Rightarrow 4x = 180 \Rightarrow x = 45
$$
✔ Answer: $ x = 45 $
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🔹 9. Ninth Figure (Bottom Right)
- Two lines intersect at O.
- One angle is $ 70^\circ $, another is $ 2x^\circ $
- These two angles are
vertically opposite → so they are equal.
$$
2x = 70 \Rightarrow x = 35
$$
✔ Answer: $ x = 35 $
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🔹 10. Tenth Figure (Bottom Left Corner)
- Angles at point O: $ 60^\circ $, $ 4x^\circ $, $ 40^\circ $
- These three angles form a straight line? Let's see.
- From the diagram: $ 60^\circ $, $ 4x $, and $ 40^\circ $ are adjacent angles on a straight line.
- So total = $ 180^\circ $
$$
60 + 4x + 40 = 180 \Rightarrow 4x + 100 = 180 \Rightarrow 4x = 80 \Rightarrow x = 20
$$
✔ Answer: $ x = 20 $
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✔ Final Answers:
| Figure | Answer |
|--------|--------|
| 1 | $ x = 120 $ |
| 2 | $ x = 36 $ |
| 3 | $ x = 85 $ |
| 4 | $ x = 45 $ |
| 5 | $ x = 35 $ |
| 6 | $ x = 25 $, $ y = 50 $ |
| 7 | $ x = 50 $ |
| 8 | $ x = 45 $ |
| 9 | $ x = 35 $ |
| 10 | $ x = 20 $ |
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📌 Key Concepts Used:
-
Linear Pair: Two adjacent angles that add up to $ 180^\circ $
-
Vertically Opposite Angles: Equal when two lines intersect
-
Angles on a Straight Line: Sum to $ 180^\circ $
-
Angles Around a Point: Sum to $ 360^\circ $
Let me know if you'd like this as a printable answer key!
Parent Tip: Review the logic above to help your child master the concept of angles and lines worksheet.
