Plane Geometry — Adjacent & Vertical Angles (Part A)
Plane Geometry — Adjacent & Vertical Angles (Part A). Practice questions to deepen understanding of adjacent and vertical angles in plane geometry. Online math practice with full solutions and step-by-step explanations.
Adjacent and Vertical Angles (Part A) — adjacent angles sum to 180°, vertical angles are equal. Diagrams and visual explanations.
1. Adjacent angles complete each other to a straight angle. 2. Vertical angles are equal to each other.
🔷 Question 1: Supplementary angles
Two supplementary angles. The first angle is 65°.
What is the second angle? 🤔
💡 Detailed explanation
📌 Theorem: Supplementary angles
Supplementary angles add up to 180°.
✨ Step-by-step solution:
Step 1️⃣: Supplementary angles add up to 180°
65° + second angle = 180°
Step 2️⃣: Solve:
Second angle = 180° − 65° = 115° ✅
🔷 Question 2: Supplementary angles
Two supplementary angles. One of them is 42°.
What is the second angle? 🧮
💡 Detailed explanation
✨ Solution:
Step 1️⃣: Supplementary angles add up to 180°
42° + x = 180°
Step 2️⃣: Solve:
x = 180° − 42° = 138°
🔶 Question 3: Vertical angles
Two lines intersect. One of the angles is 75°.
What is the vertical angle to it? 🎯
💡 Detailed explanation
📌 Theorem: Vertical angles
Vertical angles are equal to each other.
✨ Solution:
One angle = 75° → vertical angle = 75° ✅
🔷 Question 4: Supplementary angles
One angle is 90°. What is the supplementary angle to it? 📐
💡 Detailed explanation
✨ Special case — right angle:
90° + x = 180°
x = 180° − 90° = 90°
Both supplementary angles equal 90° — they are both right angles!
🔶 Question 5: Vertical angles
Two lines intersect forming an angle of 120°.
What is the vertical angle? 🎲
💡 Detailed explanation
If one angle = 120°, then the vertical angle = 120° ✅
💡 Note:
Vertical angles are always equal, regardless of the size!
🔷 Question 6: Supplementary angles
Two supplementary angles. One of them is 30°.
How many degrees is the second angle? 📏
💡 Explanation
30° + x = 180°
x = 180° − 30° = 150° ✅
🔍 Check: 30° + 150° = 180° ✓
🔶 Question 7: Vertical angles
Given an angle of size 45°. What is the vertical angle to it? 🎯 🎯
💡 Explanation
Vertical angles are equal: 45° ✅
🔷 Question 8: Supplementary angles
One angle is 110°. What is the supplementary angle? 🔍
💡 Explanation
110° + x = 180°
x = 180° − 110° = 70° ✅
🔶 Question 9: Vertical angles
At the intersection of two lines, angle one is 155°.
What is the vertical angle to it? 📊
💡 Explanation
Vertical angles are equal: 155° ✅
🔷 Question 10: Supplementary angles
One angle is 55°. What is the supplementary angle to it? 🎨
💡 Explanation
180° − 55° = 125° ✅
🔷 Question 11: Supplementary angles
Two supplementary angles. How many degrees do the two angles total? 📐
💡 Explanation
📌 Definition:
Supplementary angles are angles that always sum to 180°.
α + β = 180°
Therefore the two angles together always total 180°. ✅
🔶 Question 12: Vertical angles
When two intersecting lines, How many pairs of vertical angles are formed? 🔢
💡 Explanation
🟡 First pair: two angles α (equal!)
🟢 Second pair: two angles β (equal!)
✅ Answer: 2 pairs
🔷 Question 13: Supplementary angles
Two supplementary angles. The first angle is 3x and the second angle is 2x.
What is the value of x? 🧮 🧮
💡 Explanation
Step 1️⃣: Write the equation
3x + 2x = 180°
Step 2️⃣: Combine
5x = 180°
Step 3️⃣: Solve
x = 36° ✅
🔶 Question 14: Vertical angles
One angle at the intersection is 2x + 10°. The vertical angle to it is 70°.
What is x? 🎯
💡 Explanation
Step 1️⃣: Vertical angles are equal
2x + 10° = 70°
Step 2️⃣: Solve
2x = 70° − 10° = 60°
x = 30° ✅
🔷 Question 15: Supplementary angles
One angle is 25°. What is the supplementary angle? 📏
💡 Explanation
180° − 25° = 155° ✅
🔶 Question 16: Vertical angles
Given an angle 100°. What is the vertical angle? 🎪
Vertical angles are equal: 100° ✅
🔷 Question 17: Supplementary angles
One angle is 80°. What is the supplementary angle? 🌟
180° − 80° = 100° ✅
🔶 Question 18: Vertical angles
The angle is 35°. What is the vertical angle? 🎭
Vertical angles are equal: 35° ✅
🔷 Question 19: Supplementary angles
One angle is 135°. What is the supplementary angle? 🎨
180° − 135° = 45° ✅
🔶 Question 20: Vertical angles
The angle is 130°. What is the vertical angle? 🎯
130° ✅
Q21: Combined
Two lines intersect. One angle is 65°.
How many degrees is the supplementary angle?
Explanation: Supplementary to 65°: 180−65=115°
Q22: Supplementary angles
Two supplementary angles: x+20° and 2x−10°.
Find x.
Explanation: (x+20)+(2x−10)=180 → 3x+10=180 → 3x=170 → x≈56.67°
Q23: Vertical angles
One angle is 3x and its vertical angle is x+60°.
Find x.
Explanation: Vertical angles equal: 3x=x+60 → 2x=60 → x=30°
Q24: Combined
Two lines intersect. One angle is 48°.
What is the sum of the supplementary angle and the vertical angle?
Explanation: Supplementary=132°, vertical=48°. Sum=132+48=180°
Q25: Ratio of angles
Two supplementary angles in ratio 2:3.
What is the smaller angle?
Explanation: Let 2k+3k=180 → k=36. Smaller=2×36=72°
Q26: Combined
An angle is x. Its supplementary angle is 40° greater than it.
Find x.
Explanation: x+(x+40)=180 → 2x=140 → x=70°
Q27: Vertical angles
First angle is 5x−20° and its vertical angle is 3x+10°.
Find the first angle.
Explanation: 5x−20=3x+10 → 2x=30 → x=15. Angle=5(15)−20=55°
Q28: Three angles on a line
Three angles on a straight line: 50°, 70°, and a third.
What is the third?
Explanation: Three angles on a straight line sum to 180°. 180−50−70=60°
Q29: Four angles
Two lines intersect. Two angles are 80° each.
How many degrees is each of the other two angles?
Explanation: Adjacent angles are supplementary: 180−80=100°
Q30: Ratio
Two supplementary angles in ratio 5:7.
What is the larger angle?
Explanation: 5k+7k=180 → k=15. Larger=7×15=105°
Q31: Equation
Supplementary angles: 2x+15° and 3x−5°.
What is the larger angle?
Explanation: (2x+15)+(3x−5)=180 → 5x+10=180 → x=34. Larger=3(34)−5=97°
Q32: Vertical angles
The sum of two vertical angles is 160°.
What is each angle?
Explanation: Vertical angles are equal, so each = 160÷2=80°
Q33: Difference
Two supplementary angles differ by 26°.
What is the smaller angle?
Explanation: x+(x+26)=180 → 2x=154 → x=77°
Q34: Equation
Supplementary angles: 4x and x+90°.
Find x.
Explanation: 4x+(x+90)=180 → 5x=90 → x=18°
Q35: Ratio
Supplementary angles in ratio 4:5.
What is the difference?
Explanation: 4k+5k=180 → k=20. Angles: 80° and 100°. Difference=20°
Q36: Vertical angles
Angle 6x−15° and vertical angle 4x+25°.
What is the angle?
Explanation: 6x−15=4x+25 → 2x=40 → x=20. Angle=6(20)−15=105°
Q37: Simple ratio
One angle is twice its supplementary angle.
What is the smaller angle?
Explanation: Let smaller=x, larger=2x. x+2x=180 → x=60°
Q38: Combined
An angle is x. The supplementary is 50° larger.
What is the vertical angle of the larger angle?
Explanation: x+(x+50)=180 → x=65°. Larger=115°. Vertical=115°
Q39: Ratio
Supplementary angles: x and 2x. Ratio 1:2.
What is the larger angle?
Explanation: x+2x=180 → x=60. Larger=120°
Q40: Summary
Supplementary angles: 3x+10° and 5x−30°.
What is the sum of their vertical angles?
Explanation: Supplementary angles sum to 180°. Their verticals are equal to them, so sum of verticals = 180°.