Chapter 6, Problem 7RP

Solution

a.

  $\overline{AD}$ and m are perpendicular or not

They are perpendicular to each other

Given: $\angle ADC={\left(x+88\right)}^{\circ }$ , $\angle ADE={\left(74-8x\right)}^{\circ }$ and $\angle BDE={\left(2x+94\right)}^{\circ }$

Explanation: $\overline{AD}$ and m are perpendicular to each other because $\overline{AD}$ stands on plane m.

Conclusion: Hence, they are perpendicular to each other.

b.

  $\overline{AD}$ and $\overline{CD}$ are perpendicular or not.

  $\overline{AD}$ and $\overline{CD}$ are not perpendicular.

Given: $\angle ADC={\left(x+88\right)}^{\circ }$ , $\angle ADE={\left(74-8x\right)}^{\circ }$ and $\angle BDE={\left(2x+94\right)}^{\circ }$

Concept used: Here, we are using the concept of linear angle theorem

Calculation: Here, $\angle ADE+\angle BDE={180}^{\circ }$ (linear property)

➔ $\left(74-8x^{\circ }\right)+\left(2x+{94}^{\circ }\right)={180}^{\circ }$

➔ ${\left(-6x+168\right)}^{\circ }={180}^{\circ }$

➔ $-6x={12}^{\circ }$

➔ $x=-2$

And, $\angle ADC=x+{88}^{\circ }$

➔ $-2+{88}^{\circ }={86}^{\circ }$

Conclusion: Hence, $\overline{AD}$ and $\overline{CD}$ are not perpendicular.

c.

  $\overline{AD}$ and $\overline{DE}$ are perpendicular or not.

  $\overline{AD}$ and $\overline{DE}$ are perpendicular to each other.

Given: $\angle ADC={\left(x+88\right)}^{\circ }$ , $\angle ADE={\left(74-8x\right)}^{\circ }$ and $\angle BDE={\left(2x+94\right)}^{\circ }$

Concept used: Here, we are using the concept of linear angle theorem

Calculation: Here, $\angle ADE={\left(74-8x\right)}^{\circ }(\because$$x=-2$ )

➔ $74+16={90}^{\circ }$

Conclusion: Hence, $\overline{AD}$ and $\overline{DE}$ are perpendicular to each other.