Explanation:
We are given 8 cyclic quadrilaterals (quadrilaterals inscribed in a circle). For any cyclic quadrilateral, the
sum of opposite angles is 180°. That’s the key rule we’ll use.
Let’s solve each one step by step:
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Problem 1: Angles: 82°, 80°, x°, y°
Opposite angles sum to 180°.
Assume 82° and y° are opposite → y = 180 − 82 = 98°
Assume 80° and x° are opposite → x = 180 − 80 = 100°
✔ Check: 82 + 98 = 180, 80 + 100 = 180
So:
x = 100, y = 98
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Problem 2: Angles: 84°, x°, y°, 75°
Opposite pairs: 84° ↔ y°, x° ↔ 75°
→ y = 180 − 84 = 96°
→ x = 180 − 75 = 105°
✔ Check: 84 + 96 = 180, 105 + 75 = 180
So:
x = 105, y = 96
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Problem 3: Angles: x°, y°, 55°, 78°
Opposite: x ↔ 78, y ↔ 55
→ x = 180 − 78 = 102°
→ y = 180 − 55 = 125°
✔ Check: 102 + 78 = 180, 125 + 55 = 180
So:
x = 102, y = 125
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Problem 4: Angles: x°, 88°, 108°, y°
Opposite: x ↔ 108, 88 ↔ y
→ x = 180 − 108 = 72°
→ y = 180 − 88 = 92°
✔ Check: 72 + 108 = 180, 88 + 92 = 180
So:
x = 72, y = 92
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Problem 5: Angles: x°, ?, 67°, ?
Wait — only two angles shown: x° and 67°. But it's a quadrilateral inscribed in a circle, so the other two angles must be opposite to these. Since only two are labeled, likely x and 67 are
opposite, because they’re on opposite corners (based on diagram layout). So:
→ x + 67 = 180 → x = 113°
(Other two angles not asked; only x is missing.)
So:
x = 113
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Problem 6: Angles: 47°, x°, 63°, y°
Opposite: 47 ↔ y, 63 ↔ x
→ y = 180 − 47 = 133°
→ x = 180 − 63 = 117°
✔ Check: 47 + 133 = 180, 63 + 117 = 180
So:
x = 117, y = 133
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Problem 7: Angles: 68°, 78°, (x − 10)°, (y + 30)°
Opposite pairs: assume 68 ↔ (y + 30), and 78 ↔ (x − 10)
Then:
68 + (y + 30) = 180 → y + 98 = 180 → y = 82
78 + (x − 10) = 180 → x + 68 = 180 → x = 112
✔ Check: 68 + (82 + 30) = 68 + 112 = 180 ✔
78 + (112 − 10) = 78 + 102 = 180 ✔
So:
x = 112, y = 82
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Problem 8: Angles: 105°, (4y + 14)°, (7y + 1)°, (7x + 5)°
We need to identify opposite angles. Based on typical layout (top-left = 105°, top-right = 4y+14, bottom-right = 7x+5, bottom-left = 7y+1), likely:
- 105° opposite (7x + 5)°
- (4y + 14)° opposite (7y + 1)°
So set up equations:
1) 105 + (7x + 5) = 180
→ 7x + 110 = 180
→ 7x = 70
→ x = 10
2) (4y + 14) + (7y + 1) = 180
→ 11y + 15 = 180
→ 11y = 165
→ y = 15
✔ Check:
105 + (7·10 + 5) = 105 + 75 = 180 ✔
(4·15 + 14) = 60 + 14 = 74
(7·15 + 1) = 105 + 1 = 106
74 + 106 = 180 ✔
So:
x = 10, y = 15
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Now compile all answers:
1. x = 100, y = 98
2. x = 105, y = 96
3. x = 102, y = 125
4. x = 72, y = 92
5. x = 113
6. x = 117, y = 133
7. x = 112, y = 82
8. x = 10, y = 15
Final Answer:
1. x=100, y=98
2. x=105, y=96
3. x=102, y=125
4. x=72, y=92
5. x=113
6. x=117, y=133
7. x=112, y=82
8. x=10, y=15
Parent Tip: Review the logic above to help your child master the concept of quadrilaterals angles worksheet.
