In the rhombus, mAn angle symbol is shown.1 = 18x, mAn angle symbol is shown.2 = x + y, and mAn angle symbol is shown.3 = 30z. Find the value of x + y + z. The diagram is not drawn to scale.
A quadrilateral is shown with its diagonals and 4 congruent sides. At the intersection of the diagonals, the upper left angle is labeled 3. The lower left angle is labeled 2, and the lower right angle is labeled 1.
(1 point)
Responses
93
93
85
85
107
107
79
To solve this problem, we need to set up a system of equations using the given information.
Since the rhombus has 4 congruent sides, the measures of angles 1, 2, and 3 must add up to 360 degrees (since the sum of the angles in a quadrilateral is 360 degrees).
So we have: 18x + x + y + 30z = 360.
Since angles 1 and 2 are adjacent angles in a rhombus, they are supplementary. Therefore, we have: 18x + x + y = 180.
Simplifying this equation gives us: 19x + y = 180.
We do not have enough information to solve for x and y, so we need to use the third equation.
Since angles 2 and 3 are also adjacent angles in a rhombus, they are supplementary. Therefore, we have: x + y + 30z = 180.
Simplifying this equation gives us: x + y + 30z = 180.
Now we have a system of two equations:
19x + y = 180
x + y + 30z = 180
We can solve this system by substitution or elimination.
First, let's solve for y in the first equation:
y = 180 - 19x
Now we can substitute this expression for y in the second equation:
x + (180 - 19x) + 30z = 180
Simplifying this equation gives us:
180 - 18x + 30z = 180
Subtracting 180 from both sides gives us:
-18x + 30z = 0
Dividing both sides by 6 gives us:
-3x + 5z = 0
Now we need to rewrite the equation in terms of x + y + z.
From the first equation, we know that:
y = 180 - 19x
Substituting this expression for y in x + y + 30z = 180 gives us:
x + (180 - 19x) + 30z = 180
Simplifying this equation gives us:
180 - 18x + 30z = 180
Subtracting 180 from both sides gives us:
-18x + 30z = 0
Dividing both sides by 3 gives us:
-3x + 5z = 0
So x + y + z = x + (180 - 19x) + z = 180 + z.
Since we do not have enough information to solve for x, y, and z, we cannot determine the value of x + y + z.