if the measurement of angle ABC is one degree less than three times the measurement of angle ABD and the measurement of angle DBC= 47 degrees, find each measure

Let's define the angles to solve the problem:

- Let \( m \angle ABC \) be the measurement of angle ABC.
- Let \( m \angle ABD \) be the measurement of angle ABD.
- Let \( m \angle DBC = 47^\circ \), which is given.

According to the problem:

1. \( m \angle ABC \) is one degree less than three times \( m \angle ABD \).
   \[
   m \angle ABC = 3 \times m \angle ABD - 1
   \]
   
2. \( m \angle ABC = m \angle ABD + m \angle DBC \) since \( \angle ABC \) is the sum of angles \( \angle ABD \) and \( \angle DBC \).

Now, let's use these relationships to find the measurements of the angles.

Substitute the expression for \( m \angle ABC \) from the first equation into the second equation:
\[
3 \times m \angle ABD - 1 = m \angle ABD + 47^\circ
\]

Now, solve for \( m \angle ABD \):

\[
3 \times m \angle ABD - m \angle ABD = 47^\circ + 1
\]

\[
2 \times m \angle ABD = 48^\circ
\]

\[
m \angle ABD = \frac{48^\circ}{2} = 24^\circ
\]

Now that we know \( m \angle ABD = 24^\circ \), we can find \( m \angle ABC \):

\[
m \angle ABC = 3 \times 24^\circ - 1 = 72^\circ - 1 = 71^\circ
\]

So, the measurements of the angles are:

- \( m \angle ABD = 24^\circ \)
- \( m \angle ABC = 71^\circ \)
- \( m \angle DBC = 47^\circ \) (as given)

Would you like further details on this solution or have any additional questions?

Here are five related questions you might find interesting:

1. What is the sum of angles in a triangle?
2. How do you solve for an unknown angle in a triangle given the other two angles?
3. Can you solve for the angles in a right triangle given one angle?
4. What are supplementary angles, and how do they relate to each other?
5. How can the Law of Sines be applied to solve triangles?

**Tip:** When solving angle problems, always consider whether the angles are supplementary, complementary, or part of a triangle, as these relationships often provide additional useful equations.

Learn how to solve a problem involving angle measurements where angle ABC is one degree less than three times angle ABD, and angle DBC is given as 47 degrees. Follow a step-by-step solution method, including substitution and equation solving. Suitable for students in Grades 7-9 studying geometry and angle properties.

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