Asked by loki
The measures of the angles of a triangle are shown in the figure below. Find the measure of the smallest angle. (6x+3)° (2x+7)° 90° Answer
Answers
Answered by
GPT-4o mini
To find the measure of the smallest angle in the triangle, we first need to establish an equation using the fact that the sum of the angles in a triangle is \(180^\circ\).
Given the angles are:
- \( (6x+3)^\circ \)
- \( (2x+7)^\circ \)
- \( 90^\circ \)
We can write the equation:
\[
(6x + 3) + (2x + 7) + 90 = 180
\]
Now, combine like terms:
\[
6x + 2x + 3 + 7 + 90 = 180
\]
This simplifies to:
\[
8x + 100 = 180
\]
Next, we solve for \(x\):
\[
8x = 180 - 100
\]
\[
8x = 80
\]
\[
x = 10
\]
Now that we have \(x\), we can find the measures of the angles:
1. \( (6x + 3)^\circ = 6(10) + 3 = 60 + 3 = 63^\circ \)
2. \( (2x + 7)^\circ = 2(10) + 7 = 20 + 7 = 27^\circ \)
3. \( 90^\circ \)
Now we compare the angles \( 63^\circ \), \( 27^\circ \), and \( 90^\circ \) to determine the smallest angle.
The smallest angle is:
\[
\boxed{27^\circ}
\]
Given the angles are:
- \( (6x+3)^\circ \)
- \( (2x+7)^\circ \)
- \( 90^\circ \)
We can write the equation:
\[
(6x + 3) + (2x + 7) + 90 = 180
\]
Now, combine like terms:
\[
6x + 2x + 3 + 7 + 90 = 180
\]
This simplifies to:
\[
8x + 100 = 180
\]
Next, we solve for \(x\):
\[
8x = 180 - 100
\]
\[
8x = 80
\]
\[
x = 10
\]
Now that we have \(x\), we can find the measures of the angles:
1. \( (6x + 3)^\circ = 6(10) + 3 = 60 + 3 = 63^\circ \)
2. \( (2x + 7)^\circ = 2(10) + 7 = 20 + 7 = 27^\circ \)
3. \( 90^\circ \)
Now we compare the angles \( 63^\circ \), \( 27^\circ \), and \( 90^\circ \) to determine the smallest angle.
The smallest angle is:
\[
\boxed{27^\circ}
\]
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