Problem Analysis:
The problem involves using the
Angle Addition Postulate to solve for an unknown variable \( x \). The Angle Addition Postulate states that if a point lies in the interior of an angle, the measure of the entire angle is equal to the sum of the measures of the two smaller angles formed.
From the image:
- The angle \( \angle ADC \) is divided into two smaller angles by point \( B \):
- One angle is labeled as \( 2x \).
- The other angle is labeled as \( 5x - 1 \).
- The total measure of \( \angle ADC \) is given as \( 115^\circ \).
We need to find the value of \( x \).
---
Step-by-Step Solution:
1.
Identify the given information:
- The measure of \( \angle ADB \) is \( 2x \).
- The measure of \( \angle BDC \) is \( 5x - 1 \).
- The total measure of \( \angle ADC \) is \( 115^\circ \).
2.
Apply the Angle Addition Postulate:
According to the postulate, the measure of the entire angle \( \angle ADC \) is the sum of the measures of the two smaller angles \( \angle ADB \) and \( \angle BDC \):
\[
m\angle ADC = m\angle ADB + m\angle BDC
\]
3.
Substitute the given values:
\[
115 = 2x + (5x - 1)
\]
4.
Simplify the equation:
Combine like terms on the right-hand side:
\[
115 = 2x + 5x - 1
\]
\[
115 = 7x - 1
\]
5.
Isolate the term with \( x \):
Add 1 to both sides of the equation:
\[
115 + 1 = 7x
\]
\[
116 = 7x
\]
6.
Solve for \( x \):
Divide both sides by 7:
\[
x = \frac{116}{7}
\]
\[
x = 16.5714 \quad \text{(approximately)}
\]
However, since the problem likely expects an exact value, we keep it as a fraction:
\[
x = \frac{116}{7}
\]
---
Final Answer:
\[
\boxed{\frac{116}{7}}
\]
Parent Tip: Review the logic above to help your child master the concept of geometry angle addition worksheet.
