Question
2 horizontal and parallel lines are intersected by 2 diagonal lines to form a triangle with exterior angles. The top angle of the triangle is (2 x + 10) degrees. The bottom right angle of the triangle is 79 degrees. The exterior angle to the bottom left angle is (6 x + 1) degrees.
What is the value of x?
x = 2.25
x = 11.25
x = 13
x = 22
What is the value of x?
x = 2.25
x = 11.25
x = 13
x = 22
Answers
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Answered by
GPT-4o mini
To solve for \( x \) in the given geometry problem, we can use the properties of triangles and exterior angles.
1. The angles of a triangle sum up to \( 180^\circ \). We know the following angles:
- Top angle: \( (2x + 10)^\circ \)
- Bottom right angle: \( 79^\circ \)
- Let the bottom left angle be \( y \).
Using the triangle angle sum property:
\[
(2x + 10) + 79 + y = 180
\]
This simplifies to:
\[
2x + y + 89 = 180
\]
Subtract \( 89 \) from both sides:
\[
2x + y = 91 \tag{1}
\]
2. The exterior angle corresponding to the bottom left angle \( y \) is given as \( (6x + 1)^\circ \). The exterior angle of a triangle is equal to the sum of the two opposite interior angles, which means:
\[
(6x + 1) = (2x + 10) + 79
\]
This can be rearranged to:
\[
(6x + 1) = 2x + 89
\]
Subtracting \( 2x \) from both sides gives:
\[
6x - 2x + 1 = 89
\]
This simplifies to:
\[
4x + 1 = 89
\]
Subtracting \( 1 \) from both sides:
\[
4x = 88
\]
Dividing both sides by \( 4 \):
\[
x = 22
\]
Having solved for \( x \), we can now verify our solution:
1. If \( x = 22 \):
- The top angle is \( 2(22) + 10 = 44 + 10 = 54^\circ \).
- The bottom left angle can be found using \( 2x + y = 91 \):
- \( 2(22) + y = 91 \) implies \( 44 + y = 91 \), which means \( y = 91 - 44 = 47^\circ \).
- Now, checking the exterior angle:
- Bottom left angle \( y = 47^\circ \).
- The exterior angle to the bottom left angle is \( (6(22) + 1) = 132 + 1 = 133^\circ \).
- The sum of the interior angles for the triangle:
- \( 54 + 79 + 47 = 180 \) (which is correct).
- The exterior angle \( 133^\circ \) is equal to the sum \( 54 + 79 = 133 \) (also correct).
Thus, the value of \( x \) is
\[
\boxed{22}
\]
1. The angles of a triangle sum up to \( 180^\circ \). We know the following angles:
- Top angle: \( (2x + 10)^\circ \)
- Bottom right angle: \( 79^\circ \)
- Let the bottom left angle be \( y \).
Using the triangle angle sum property:
\[
(2x + 10) + 79 + y = 180
\]
This simplifies to:
\[
2x + y + 89 = 180
\]
Subtract \( 89 \) from both sides:
\[
2x + y = 91 \tag{1}
\]
2. The exterior angle corresponding to the bottom left angle \( y \) is given as \( (6x + 1)^\circ \). The exterior angle of a triangle is equal to the sum of the two opposite interior angles, which means:
\[
(6x + 1) = (2x + 10) + 79
\]
This can be rearranged to:
\[
(6x + 1) = 2x + 89
\]
Subtracting \( 2x \) from both sides gives:
\[
6x - 2x + 1 = 89
\]
This simplifies to:
\[
4x + 1 = 89
\]
Subtracting \( 1 \) from both sides:
\[
4x = 88
\]
Dividing both sides by \( 4 \):
\[
x = 22
\]
Having solved for \( x \), we can now verify our solution:
1. If \( x = 22 \):
- The top angle is \( 2(22) + 10 = 44 + 10 = 54^\circ \).
- The bottom left angle can be found using \( 2x + y = 91 \):
- \( 2(22) + y = 91 \) implies \( 44 + y = 91 \), which means \( y = 91 - 44 = 47^\circ \).
- Now, checking the exterior angle:
- Bottom left angle \( y = 47^\circ \).
- The exterior angle to the bottom left angle is \( (6(22) + 1) = 132 + 1 = 133^\circ \).
- The sum of the interior angles for the triangle:
- \( 54 + 79 + 47 = 180 \) (which is correct).
- The exterior angle \( 133^\circ \) is equal to the sum \( 54 + 79 = 133 \) (also correct).
Thus, the value of \( x \) is
\[
\boxed{22}
\]
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