Question
The measures of the angles in a triangle are given. - The measure of Angle \( A \) is \( (x+6) \) - The measure of Angle \( B \) is twice the measure of Angle \( A \). - The measure of Angle \( C \) is equal to measure of Angle B. What is the value of \( x \) ? A. 30 B. 35 C. 45 D. 50
Ask by Ellis Delgado. in the United States
Jan 31,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 30 \)
Solution
Let's solve for \( x \) using the properties of a triangle:
1. **Given Angles:**
- Angle \( A = x + 6 \)
- Angle \( B = 2A = 2(x + 6) \)
- Angle \( C = B = 2(x + 6) \)
2. **Sum of Angles in a Triangle:**
\[
A + B + C = 180^\circ
\]
Substitute the given angles:
\[
(x + 6) + 2(x + 6) + 2(x + 6) = 180
\]
3. **Simplify the Equation:**
\[
x + 6 + 2x + 12 + 2x + 12 = 180
\]
Combine like terms:
\[
5x + 30 = 180
\]
4. **Solve for \( x \):**
\[
5x = 180 - 30
\]
\[
5x = 150
\]
\[
x = \frac{150}{5} = 30
\]
So, the value of \( x \) is **30**.
**Answer:** A. 30
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Simplify this solution Extra Insights
To find the value of \( x \), we first express all angles in terms of \( x \). We have: - Angle \( A = x + 6 \) - Angle \( B = 2A = 2(x + 6) = 2x + 12 \) - Angle \( C = B = 2x + 12 \) Now, we can use the triangle angle sum property, which states that the sum of the angles in a triangle is always \( 180^\circ \): \[ A + B + C = 180 \] Substituting in the values, we get: \[ (x + 6) + (2x + 12) + (2x + 12) = 180 \] This simplifies to: \[ x + 6 + 2x + 12 + 2x + 12 = 180 \] Combining like terms: \[ 5x + 30 = 180 \] Now, we can solve for \( x \): \[ 5x = 180 - 30 \] \[ 5x = 150 \] \[ x = 30 \] So the value of \( x \) is \( 30\) and the correct answer is A.
