Question

Each figure in Exercises 19-24 is a parallelogram with its diagonals drawn. Find the values of x and y.

Short answer

The value of x and y is 13 and 5 respectively.

Step-by-step solution

Step 1. Apply property of parallelogram.

Opposite angles of a parallelogram are congruent, that is they are equal on measure.

In a given figure, by the property of a parallelogram,

$\begin{array}{c}m\angle B=m\angle D\\ =80°\end{array}$

$\begin{array}{c}m\angle A=m\angle C\\ 5x+6y+5=35+m\angle ACB\end{array}$

Step 2. Description of a step.

The sum of measures of all the angles of a triangle is 180.

Consider $\Delta ADC$, such that,

$\begin{array}{c}m\angle ADC+m\angle DCA+m\angle DAC=180\\ 80+35+5x=180\\ 115+5x=180\\ 5x=65\end{array}$

Step 3. Description of a step.

Divide each side of $5x=65$ by 5 to obtain the value of x.

$\begin{array}{c}\frac{5x}{5}=\frac{65}{5}\\ x=13\end{array}$

Step 4. Description of a step.

The sum of measures of all the angles of a triangle is 180.

Consider $\Delta ABC$, such that,

$\begin{array}{c}m\angle ACB+m\angle ABC+m\angle BAC=180\\ m\angle ACB+80+6y+5=180\\ m\angle ACB+6y=95\\ m\angle ACB=95-6y\end{array}$

Step 5. Description of a step.

Substitute $95-6y$for $m\angle ACB$and 13 for into $5x+6y+5=35+m\angle ACB$ to find the value of y.

$\begin{array}{c}5x+6y+5=35+95-6y\\ 5\times 13+6y+5=35+95-6y\\ 12y=60\\ y=5\end{array}$

Therefore, the value of x and y is 13 and 5 respectively.