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Consider the figure.

An illustration shows three rays A, D, and C emerging from a point labeled B, forming a right angle at B. The right angle is divided into two parts by ray B D. The angles face leftward. Angle A B C measures 90 degree, angle A B D measures 4 x degrees, and angle D B C measures left parenthesis 7 x minus 9 right parenthesis degrees.

(3 points)
x
=

7x−9
=

4x
=

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1 answer

To solve for \( x \) in the given problem, we can use the information provided about the angles.

We know the following:

  • Angle \( A B C \) measures \( 90^\circ \)
  • Angle \( A B D \) measures \( 4x \)
  • Angle \( D B C \) measures \( 7x - 9 \)

Since the angles \( A B D \) and \( D B C \) together form the right angle \( A B C \), we can set up the equation:

\[ 4x + (7x - 9) = 90 \]

Now, let's simplify and solve for \( x \):

  1. Combine like terms: \[ 4x + 7x - 9 = 90 \] \[ 11x - 9 = 90 \]

  2. Add 9 to both sides: \[ 11x = 90 + 9 \] \[ 11x = 99 \]

  3. Divide by 11: \[ x = \frac{99}{11} \] \[ x = 9 \]

With \( x = 9 \), we can now find the measures of the angles \( A B D \) and \( D B C \):

  1. Calculate \( 4x \): \[ 4x = 4(9) = 36^\circ \]

  2. Calculate \( 7x - 9 \): \[ 7x - 9 = 7(9) - 9 = 63 - 9 = 54^\circ \]

Now, we can summarize our results:

  • \( x = 9 \)
  • \( 7x - 9 = 54^\circ \)
  • \( 4x = 36^\circ \)

Thus, the final answers are:

\[ x = 9 \] \[ 7x - 9 = 54 \] \[ 4x = 36 \]