Question

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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Answer 1

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:

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