Chapter 5, Problem 18RP

a.

To determine

To Find: The value of $m\angle 1$ .

Answer to Problem 18RP

  $m\angle 1={116.82}^{\circ }$

Explanation of Solution

Given:

  

  $\begin{array}{l}\overleftrightarrow{AD}\parallel \overleftrightarrow{BC}\\ m\angle 1={\left(5.63x+2.42\right)}^{\circ }\\ m\angle 2={\left(2.1x\right)}^{\circ }\\ m\angle 3={\left(6x-5.1\right)}^{\circ }\\ m\angle 4={42}^{\circ }\end{array}$

Concept Used:

When two parallel lines intersected by a transversal, then:

• Alternate interior angles are equal.
• Corresponding angles are equal.
• Angles on the same side of transversal are supplementary.

Calculation:

  $\begin{array}{l}\overleftrightarrow{AD}\parallel \overleftrightarrow{BC}\\ m\angle 1={\left(5.63x+2.42\right)}^{\circ }\\ m\angle 2={\left(2.1x\right)}^{\circ }\\ m\angle 3={\left(6x-5.1\right)}^{\circ }\\ m\angle 4={42}^{\circ }\end{array}$

  $\begin{array}{l}\because \overleftrightarrow{AD}\parallel \overleftrightarrow{BC}\\ \Rightarrow m\angle 1=m\angle 3\left(\text{Alternate interior angles are equal}\right)\\ \Rightarrow 5.63x+2.42=6x-5.1\\ \Rightarrow 5.63x-6x=-5.1-2.42\\ \Rightarrow -0.37x=-7.52\\ \Rightarrow x=\frac{7.52}{0.37}=20.32\\ \Rightarrow m\angle 1={\left(5.63\times 20.32+2.42\right)}^{\circ }={116.82}^{\circ }\\ \therefore m\angle 1={116.82}^{\circ }\end{array}$

Conclusion:

  $m\angle 1={116.82}^{\circ }$

b.

To determine

To Find: Whether $\overleftrightarrow{DC}\parallel \overleftrightarrow{AB}$ ?

Answer to Problem 18RP

  $\overleftrightarrow{DC}\cancel{\parallel }\overleftrightarrow{AB}$

Explanation of Solution

Given:

  

  $\begin{array}{l}\overleftrightarrow{AD}\parallel \overleftrightarrow{BC}\\ m\angle 1={\left(5.63x+2.42\right)}^{\circ }\\ m\angle 2={\left(2.1x\right)}^{\circ }\\ m\angle 3={\left(6x-5.1\right)}^{\circ }\\ m\angle 4={42}^{\circ }\end{array}$

Concept Used:

When two parallel lines intersected by a transversal, then:

• Alternate interior angles are equal.
• Corresponding angles are equal.
• Angles on the same side of transversal are supplementary.

Calculation:

  $\begin{array}{l}\overleftrightarrow{AD}\parallel \overleftrightarrow{BC}\\ m\angle 1={\left(5.63x+2.42\right)}^{\circ }\\ m\angle 2={\left(2.1x\right)}^{\circ }\\ m\angle 3={\left(6x-5.1\right)}^{\circ }\\ m\angle 4={42}^{\circ }\\ x=20.32\\ \therefore m\angle 2={\left(2.1\times 20.32\right)}^{\circ }={42.672}^{\circ }\ne {42}^{\circ }\\ \Rightarrow m\angle 2\ne m\angle 4\\ \text{Alternate interior angles formed by }\overleftrightarrow{DC}\text{ and }\overleftrightarrow{AB}\text{ are not equal}\text{.}\\ \Rightarrow \overleftrightarrow{DC}\cancel{\parallel }\overleftrightarrow{AB}\end{array}$

Conclusion:

  $\overleftrightarrow{DC}\cancel{\parallel }\overleftrightarrow{AB}$