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Calculate the value of x using angles around a point and vertical angles
Mathematics
Grade 9 (Junior High School)
Question Content
Calculate the value of x. The diagram shows angles around a point: 172°, 2x, 16°, x, and the remaining angle is a vertical angle equal to x (formed by two intersecting straight lines). Note: The diagram is not drawn accurately.
Correct Answer
x = 56
Detailed Solution Steps
1
Step 1: Recall that the sum of angles around a single point is 360°. Also, recognize that the two intersecting straight lines create a pair of vertical angles, so the unlabeled angle opposite to x is also x.
2
Step 2: Set up the equation for the sum of all angles: 172 + 2x + 16 + x + x = 360
3
Step 3: Combine like terms on the left side: (172 + 16) + (2x + x + x) = 360, which simplifies to 188 + 4x = 360
4
Step 4: Subtract 188 from both sides of the equation: 4x = 360 - 188, so 4x = 172
5
Step 5: Divide both sides by 4 to solve for x: x = 172 ÷ 4 = 56
Knowledge Points Involved
1
Angles around a point
The sum of all angles formed around a single fixed point is always 360°. This rule is used to solve for unknown angles when multiple angles meet at a point, by setting up an equation where the sum of all given and unknown angles equals 360°.
2
Vertical angles (Vertically opposite angles)
When two straight lines intersect, the opposite angles (vertical angles) are equal in measure. In this problem, this rule identifies that the unlabeled angle opposite to the given x is also x, allowing us to include it in the sum of angles around the point.
3
Solving linear equations with one variable
This involves combining like terms, isolating the variable term by performing inverse operations (addition/subtraction first, then multiplication/division), and solving for the unknown value. It is a foundational algebraic skill used to find unknowns in geometry and other math topics.
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