QUESTION IMAGE
Question
(9x+14), (7x+84)°, 8x+17, 17x+3, (x+20), x°, (210-3x), (9x-14)°, (5x+86)°, (circled x=180)
Let's solve each geometric problem one by one.
1. Vertical Angles (Top - Left)
Vertical angles are equal. So we set \(9x + 14=7x + 84\)
Step 1: Subtract \(7x\) from both sides
\(9x-7x + 14=7x-7x + 84\)
\(2x+14 = 84\)
Step 2: Subtract 14 from both sides
\(2x+14 - 14=84 - 14\)
\(2x=70\)
Step 3: Divide by 2
\(x=\frac{70}{2}=35\)
2. Corresponding Angles (Top - Right)
Since the lines are parallel, corresponding angles are equal. So \(8x + 17=17x+3\)
Step 1: Subtract \(8x\) from both sides
\(8x-8x + 17=17x-8x+3\)
\(17 = 9x+3\)
Step 2: Subtract 3 from both sides
\(17 - 3=9x+3 - 3\)
\(14 = 9x\)
Step 3: Divide by 9
\(x=\frac{14}{9}\approx1.56\) (if we need a fractional form, \(x = \frac{14}{9}\))
3. Triangle Angle Sum (Bottom - Left)
The sum of angles in a triangle is \(180^{\circ}\). So \(x+(x + 20)+(210-3x)=180\)
Step 1: Combine like terms
\(x+x + 20+210-3x=180\)
\((x+x-3x)+(20 + 210)=180\)
\(-x+230 = 180\)
Step 2: Subtract 230 from both sides
\(-x+230-230=180 - 230\)
\(-x=-50\)
Step 3: Multiply by - 1
\(x = 50\) (Note: The hand - written \(x = 180\) is incorrect. The correct value is \(x = 50\))
4. Alternate Exterior Angles (Bottom - Right)
If the lines are parallel, alternate exterior angles are equal. So \(9x-14=5x + 86\)
Step 1: Subtract \(5x\) from both sides
\(9x-5x-14=5x-5x + 86\)
\(4x-14 = 86\)
Step 2: Add 14 to both sides
\(4x-14 + 14=86 + 14\)
\(4x=100\)
Step 3: Divide by 4
\(x=\frac{100}{4}=25\)
Final Answers:
- For vertical angles: \(x = 35\)
- For corresponding angles: \(x=\frac{14}{9}\) (or approximately \(1.56\))
- For triangle angle sum: \(x = 50\)
- For alternate exterior angles: \(x = 25\)
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Let's solve each geometric problem one by one.
1. Vertical Angles (Top - Left)
Vertical angles are equal. So we set \(9x + 14=7x + 84\)
Step 1: Subtract \(7x\) from both sides
\(9x-7x + 14=7x-7x + 84\)
\(2x+14 = 84\)
Step 2: Subtract 14 from both sides
\(2x+14 - 14=84 - 14\)
\(2x=70\)
Step 3: Divide by 2
\(x=\frac{70}{2}=35\)
2. Corresponding Angles (Top - Right)
Since the lines are parallel, corresponding angles are equal. So \(8x + 17=17x+3\)
Step 1: Subtract \(8x\) from both sides
\(8x-8x + 17=17x-8x+3\)
\(17 = 9x+3\)
Step 2: Subtract 3 from both sides
\(17 - 3=9x+3 - 3\)
\(14 = 9x\)
Step 3: Divide by 9
\(x=\frac{14}{9}\approx1.56\) (if we need a fractional form, \(x = \frac{14}{9}\))
3. Triangle Angle Sum (Bottom - Left)
The sum of angles in a triangle is \(180^{\circ}\). So \(x+(x + 20)+(210-3x)=180\)
Step 1: Combine like terms
\(x+x + 20+210-3x=180\)
\((x+x-3x)+(20 + 210)=180\)
\(-x+230 = 180\)
Step 2: Subtract 230 from both sides
\(-x+230-230=180 - 230\)
\(-x=-50\)
Step 3: Multiply by - 1
\(x = 50\) (Note: The hand - written \(x = 180\) is incorrect. The correct value is \(x = 50\))
4. Alternate Exterior Angles (Bottom - Right)
If the lines are parallel, alternate exterior angles are equal. So \(9x-14=5x + 86\)
Step 1: Subtract \(5x\) from both sides
\(9x-5x-14=5x-5x + 86\)
\(4x-14 = 86\)
Step 2: Add 14 to both sides
\(4x-14 + 14=86 + 14\)
\(4x=100\)
Step 3: Divide by 4
\(x=\frac{100}{4}=25\)
Final Answers:
- For vertical angles: \(x = 35\)
- For corresponding angles: \(x=\frac{14}{9}\) (or approximately \(1.56\))
- For triangle angle sum: \(x = 50\)
- For alternate exterior angles: \(x = 25\)