Chapter 4.4, Problem 11CE

To determine

To find: How Corollary 3 follows from theorem 4.1.

Answer to Problem 11CE

Corollary 3 is proved by the statement of theorem 4.1 as shown in below proof.

Explanation of Solution

Given information: Statement of theorem 4.1 as, ”If two sides of an isosceles triangle are congruent, then the angles opposite to those sides are congruent.”

Corollary 3 states that, “The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.”

Concept used: Here the perpendicular bisects the given triangle. So, show that both triangles thus formed are congruent using ASA congruency, so that by CPCT, will show the base angles as 90 degree, hence perpendicular on base.

Calculation: In below isosceles triangle, using statement of theorem 4.4,

  

  $\because AB=AC\Rightarrow \angle B=\angle C$

Now in triangle $\Delta ABD$ and $\Delta ACD$ ,

  $\because AB=AC\Rightarrow \angle B=\angle C$

1. Given

  $m\angle BAD=m\angle CAD\text{Given}$

So, by ASA congruency, both the triangles are congruent. Thus, by CPCT,

  $m\angle BDA=m\angle CDA$

Again by definition of linear pair,

  $\begin{array}{l}m\angle BDA+m\angle CDA=180\\ \Rightarrow 2\angle BDA=180\\ \Rightarrow \angle BDA=90\end{array}$

It shows that $AD$ is perpendicular on base $BC$ .

Conclusion: Thus, by above steps, it is cleared that in an isosceles triangle, the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint, that is given corollary 3 that is obtained by the following statement of theorem 4.1 that if two sides of an isosceles triangle are congruent, then the angles opposite to those sides are congruent.”