Chapter 4.4, Problem 24WE

a)

To determine

To find:In the given figure, the measureof $\angle ABC$ .

Answer to Problem 24WE

The measure of $\angle ABC$ is 90 degrees.

Explanation of Solution

Giveninformation:In the given figure, $m\angle 1=35$ .

  

Conceptused: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 an isosceles triangle, $\Delta ABD$ as $BD=AD$ , so,

  $m\angle 1\cong m\angle 2=35$

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

  $m\angle 3=m\angle 1+m\angle 2=35+35=70$

In isosceles triangle, $\Delta BCD$ as $BD=CD$ , so, their opposite angles are also equal,

  $\Rightarrow m\angle 4\cong m\angle 5$

Now in triangle $\Delta BCD$ , the sum of all three angles is 180 degrees, so,

  $\begin{array}{l}\angle 3+m\angle 4+m\angle 5=180\\ \Rightarrow 70+m\angle 4+m\angle 5=180\\ \Rightarrow m\angle 5+m\angle 5=180-70=110\\ \Rightarrow 2m\angle 5=110\\ \Rightarrow m\angle 5=\frac{110}{2}=55\end{array}$

So, $m\angle ABC=m\angle 2+m\angle 5=35+55=90$

Conclusion: Hence, the measure of $\angle ABC$ is 90 degrees.

b)

To determine

To find: In the given figure, the measureof $\angle ABC$ .

Answer to Problem 24WE

The measure of angle $\angle ABC$ is 90 degree.

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, $\Delta ABD$ as $BD=AD$ , so,

  $m\angle 1\cong m\angle 2=x$

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

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

In isosceles triangle, $\Delta BCD$ as $BD=CD$ , so, their opposite angles are also equal,

  $\Rightarrow m\angle 4\cong m\angle 5$

Now in triangle $\Delta BCD$ , as the sum of all three angles is 180 degrees, so,

  $\begin{array}{l}m\angle 3+m\angle 4+m\angle 5=180\\ \Rightarrow 2x+m\angle 4+m\angle 5=180\\ \Rightarrow m\angle 5+m\angle 5=180-2x\\ \Rightarrow 2m\angle 5=180-2x\\ $ \Rightarrow m\angle 5=\frac{180-2x}{2}={(90-x)}^0$\end{array}$

So, $m\angle ABC=m\angle 2+m\angle 5=x+\left(90-x\right)=90$

Conclusion: Hence, the measure of $\angle ABC$ is 90 degrees.