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Solve for x: Find the Unknown Angle of a Triangle Using the Angle Sum Theorem
Mathematics
Grade 8 (Junior High School)
Question Content
The measures of the angles of a triangle are shown in the figure below. Solve for x. The angles are x°, 67°, and (4x+3)°.
Correct Answer
22
Detailed Solution Steps
1
Step 1: Recall the Triangle Angle Sum Theorem, which states that the sum of the interior angles of any triangle is 180°. Set up an equation using the given angles: x + 67 + (4x + 3) = 180
2
Step 2: Combine like terms on the left side of the equation. Add the variable terms: x + 4x = 5x. Add the constant terms: 67 + 3 = 70. The equation simplifies to 5x + 70 = 180
3
Step 3: Isolate the term with x by subtracting 70 from both sides of the equation: 5x + 70 - 70 = 180 - 70, which simplifies to 5x = 110
4
Step 4: Solve for x by dividing both sides of the equation by 5: 5x ÷ 5 = 110 ÷ 5, so x = 22
Knowledge Points Involved
1
Triangle Angle Sum Theorem
This theorem states that the sum of the three interior angles of any triangle is always 180 degrees. It is a fundamental property used to solve for unknown angles in triangles, applicable to all types of triangles (acute, obtuse, right, equilateral, isosceles, scalene).
2
Combining Like Terms
Like terms are terms that have the same variables raised to the same powers. When solving algebraic equations, we combine these terms by adding or subtracting their coefficients to simplify the equation, making it easier to isolate the unknown variable.
3
Solving One-Variable Linear Equations
A one-variable linear equation has the form ax + b = c (where a, b, c are constants and a ≠ 0). The process involves isolating the variable by performing inverse operations (subtraction/addition first, then multiplication/division) on both sides of the equation to maintain equality.
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