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As shown in the figure below, points A, B, and D lie on a line. The measure of angle ABC (m∠ABC) is x°, and m∠CBD is (5x+4)°.

Question

Question

As shown in the figure below, points A, B, and D lie on a line. The measure of angle ABC (mABC m∠ABC ) is , and mCBD m∠CBD is (5x + 4)°.

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Solution

1. Break Down the Problem

We need to determine the relationship between the angles given that points A, B, and D are collinear, with angle ABC measuring xx^\circ and angle CBD measuring (5x+4) (5x + 4)^\circ . Since these angles are on a straight line, we know that:

mABC+mCBD=180 m∠ABC + m∠CBD = 180^\circ

2. Relevant Concepts

The angles along a straight line add up to 180180^\circ. Therefore, we can set up the following equation based on the information given:

x+(5x+4)=180 x + (5x + 4) = 180

3. Analysis and Detail

Now, we will simplify and solve the equation:

  1. Combine like terms: x+5x+4=180    6x+4=180 x + 5x + 4 = 180 \implies 6x + 4 = 180

  2. Subtract 4 from both sides: 6x=1804    6x=176 6x = 180 - 4 \implies 6x = 176

  3. Divide by 6: x=1766    x=88329.33 x = \frac{176}{6} \implies x = \frac{88}{3} \approx 29.33

4. Verify and Summarize

Now we will verify our result by calculating mCBDm∠CBD:

mCBD=5(883)+4=4403+123=4523150.67 m∠CBD = 5 \left(\frac{88}{3}\right) + 4 = \frac{440}{3} + \frac{12}{3} = \frac{452}{3} \approx 150.67

Now, check if the angles add up to 180180^\circ:

mABC+mCBD=883+4523=5403=180 m∠ABC + m∠CBD = \frac{88}{3} + \frac{452}{3} = \frac{540}{3} = 180

Final Answer

Thus, the value of xx is 883 \frac{88}{3} or approximately 29.3329.33^\circ.

This problem has been solved

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