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To solve for \( x \) such that \(\angle BGD\) is a right angle in the given figure, we need to use the provided angle measures:

- \( m\angle BGC = (16x - 4)^\circ \)
- \( m\angle CGD = (2x + 13)^\circ \)

Since \(\angle BGD\) is a right angle, we know that:
\[
m\angle BGD = 90^\circ
\]

Also, note that:
\[
m\angle BGD = m\angle BGC + m\angle CGD
\]

Substituting the given expressions for the angles, we get:
\[
90^\circ = (16x - 4)^\circ + (2x + 13)^\circ
\]

Simplify the equation:
\[
90 = 16x - 4 + 2x + 13
\]
\[
90 = 18x + 9
\]

Subtract 9 from both sides:
\[
81 = 18x
\]

Divide both sides by 18:
\[
x = \frac{81}{18}
\]
\[
x = 4.5
\]

Thus, the value of \( x \) that makes \(\angle BGD\) a right angle is \( 4.5 \).

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:
1. What are the measures of \(\angle BGC\) and \(\angle CGD\) when \( x = 4.5 \)?
2. How would the solution change if \(\angle BGD\) were instead \(45^\circ\)?
3. If another angle, say \(\angle AGD\), was given as \( (5x + 10)^\circ \), how would you find \( x \) if \( \angle AGD \) is also a right angle?
4. How can you verify that the sum of angles around point G is \( 360^\circ \)?
5. Can this method be applied to other types of polygons or angles in different figures?

**Tip:** Always double-check your final solution by substituting the value back into the original equations to ensure consistency.

Find the value of x so that angle BGD is a right angle, given m∠BGC = (16x − 4)° and m∠CGD = (2x + 13)°.

Learn how to solve for x in a geometric problem where the sum of two angles equals a right angle. The given expressions for angles BGC and CGD are used to form an equation, simplified step by step.

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