Chapter 4.4, Problem 23WE

a)

To determine

To find:In the given figure, the values of $m\angle 3,m\angle 4,m\angle 5$ .

Answer to Problem 23WE

Required values are:

  $\begin{array}{l}m\angle 3=40\\ m\angle 4=40\\ m\angle 5=60\end{array}$

Explanation of Solution

Giveninformation:In the given figure, $m\angle 1=20$ .

  

Conceptused:In an isosceles triangle, anglesopposite to equal sides are also congruent.

Also, in a triangle, exterior angle is always equal to sum of its interior angles.

Calculation: In isosceles triangle $\Delta BCD$, as $BD=CD$ , so,

  $m\angle 1\cong m\angle 2=20$

Also, as exterior angle is equal to sum of its two opposite interior angles, so,

  $m\angle 3=m\angle 1+m\angle 2=20+20=40$

And $m\angle 4=m\angle 3=40$ (In $\Delta ABD$ , angles opposite to equal sides are equal)

Nowin $\Delta ABC$ ,

  $\angle 5=\angle 4+\angle 1=40+20=60$

Conclusion: Hence,

  $\begin{array}{l}\angle 3=40\\ \angle 4=40\\ \angle 5=60\end{array}$

b)

To determine

To find: In the given figure, the values of $m\angle 3,m\angle 4,m\angle 5$ .

Answer to Problem 23WE

Required values are:

  $\begin{array}{l}m\angle 3=2x\\ m\angle 4=2x\\ m\angle 5=3x\end{array}$

Explanation of Solution

Giveninformation: In the given figure, $m\angle 1=x$ .

  

Concept used: In an isosceles triangle, angles opposite to equal sides are also congruent. Also, in a triangle, exterior angle is always equal to sum of its interior angles.

Calculation: In isosceles triangle $\Delta BCD$, as $BD=CD$ , so,

  $m\angle 1\cong m\angle 2=x$

Also, as exterior angle is equal to sum of its two opposite interior angles, so,

  $m\angle 3=m\angle 1+m\angle 2=x+x=2x$

And $m\angle 4=m\angle 3=2x$ (In $\Delta ABD$ , angles opposite to equal sides are equal)

Nowin $\Delta ABC$ ,

  $m\angle 5=\angle 4+\angle 1=2x+x=3x$

Conclusion: Hence,

  $\begin{array}{l}m\angle 3=2x\\ m\angle 4=2x\\ m\angle 5=3x\end{array}$