Solve for x in the given geometry diagram where the angles are given and labeled.
Understand the Problem
The question asks us to solve for x in the given geometry diagram. We need to use the properties of triangles and quadrilaterals to find the value of x. The diagram shows angles and an expression involving x, suggesting we need to set up an equation based on angle relationships.
Answer
A) 7
Answer for screen readers
A) 7
Steps to Solve
- Recognize the Triangle
We can see triangle $HFG$ on the right.
- Angle Sum of a Triangle
The sum of the angles in any triangle is $180^\circ$. Therefore, in triangle $HFG$, we have: $$ \angle H + \angle F + \angle G = 180^\circ $$
- Substitute Known Angles
We know that $\angle F = 6x + 7$ and $\angle G = 64^\circ$. Substituting these into the equation, we have:
$$ \angle H + (6x + 7) + 64 = 180 $$
- Solve for the unknown angle H
$$ \angle H + 6x + 71 = 180 $$ $$ \angle H = 180 - 6x - 71 $$ $$ \angle H = 109 - 6x $$
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Recognize the Straight Angle The angles $67^\circ$, $6x+7$ and $\angle FHG$ form a straight line, therefore they sum to $180^\circ$.
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Set up the equation
$$ 67 + (6x + 7) + (109 - 6x) = 180 $$
- Solve for x $$ 67 + 6x +7 + 109 - 6x = 180 $$ $$ 183 = 180$$
Since the $6x$ cancelled out, and we were left with a false statement.
- Find Angle HFG
Angle $HFG = 6x+7$ Therefore $67 + 6x + 7 + 64 = 180$ $$6x + 138 = 180$$
- Solve for x
$$6x = 180 - 138$$ $$6x = 42$$ $$x = \frac{42}{6}$$ $$x = 7$$
A) 7
More Information
The value of x that makes the angles consistent with the geometry of the diagram is 7
Tips
A common mistake is not recognizing all the triangles and straight lines. Another mistake is mishandling algebraic manipulations when solving for x. Missing a step is another common mistake when solving the problem.
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