Chapter 5: Q21 (page 170)

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

Short Answer

Expert verified

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,

$\begin{matrix} m ∠ B = m ∠ D \\ = 80 ° \end{matrix}$

$\begin{matrix} m ∠ A = m ∠ C \\ 5 x + 6 y + 5 = 35 + m ∠ A C B \end{matrix}$

Step 2. Description of a step.

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

Consider $Δ A D C$ , such that,

$\begin{matrix} m ∠ A D C + m ∠ D C A + m ∠ D A C = 180 \\ 80 + 35 + 5 x = 180 \\ 115 + 5 x = 180 \\ 5 x = 65 \end{matrix}$

Step 3. Description of a step.

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

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

Step 4. Description of a step.

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

Consider $Δ A B C$ , such that,

$\begin{matrix} m ∠ A C B + m ∠ A B C + m ∠ B A C = 180 \\ m ∠ A C B + 80 + 6 y + 5 = 180 \\ m ∠ A C B + 6 y = 95 \\ m ∠ A C B = 95 - 6 y \end{matrix}$

Step 5. Description of a step.

Substitute $95 - 6 y$ for $m ∠ A C B$ and 13 for into $5 x + 6 y + 5 = 35 + m ∠ A C B$ to find the value of y.

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

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

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

Short Answer

Step by step solution

Step 1. Apply property of parallelogram.

Step 2. Description of a step.

Step 3. Description of a step.

Step 4. Description of a step.

Step 5. Description of a step.

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