To solve this problem, we need to set up equations based on the given information and solve for each variable.
Given:
m∠1 = 18x
m∠2 = x + y
m∠3 = 30z
From the information given in the problem, we know that opposite angles in a rhombus are congruent. Therefore, we can set up the following equation:
m∠1 + m∠3 = 180°
Substituting the given values:
18x + 30z = 180
We also know that the sum of the angles in a quadrilateral is equal to 360°. Therefore, we can set up the following equation:
m∠1 + m∠2 + m∠3 + m∠4 = 360°
Since all four sides of the rhombus are congruent, all four angles are also congruent. Therefore, we can rewrite the equation as:
4(m∠1) = 360
Substituting the given value:
4(18x) = 360
Simplifying:
72x = 360
Now we can solve for x:
x = 360/72
x = 5
Substituting the value of x back into the equation 18x + 30z = 180:
18(5) + 30z = 180
90 + 30z = 180
30z = 180 - 90
30z = 90
Now we can solve for z:
z = 90/30
z = 3
Substituting the values of x and z back into the equation m∠2 = x + y:
m∠2 = 5 + y
Since m∠2 is congruent to m∠1, we can substitute the value of m∠1 (18x) into the equation:
18x = 5 + y
Substituting the value of x from earlier:
18(5) = 5 + y
90 = 5 + y
y = 90 - 5
y = 85
Therefore, the values of each variable are:
x = 5
y = 85
z = 3
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
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.
In the rhombus, m∠1=18x
, m∠2=x+y
, and m∠3=30z
. Find the value of each variable. The diagram is not drawn to scale.
(3 points)
1 answer