Chapter 4: Q18. (page 138)

Write proofs in two–column form.
Given: $\bar{XY} ≅ \bar{XZ};$
$\bar{YO}$ bisects $∠ XYZ$
$\bar{ZO}$ bisects $∠ XZY$
Prove: $\bar{YO} ≅ \bar{ZO}$

Expert verified

Statement	Reason
$\bar{XY} ≅ \bar{XZ}$	Given
$m ∠ 1 = m ∠ 2$	bisects
$m ∠ 3 = m ∠ 4$	bisects
$m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$	Isosceles triangle theorem
$m ∠ 2 = m ∠ 3$	Since
$\bar{YO} ≅ \bar{ZO}$	Converse isosceles triangle theorem

Step by step solution

Angle bisector divides an angle into two equal parts.

From the figure in step 1 it can be observed that $\bar{Y O}$ bisects $∠ X Y Z$ and $\bar{Z O}$ bisects $∠ X Z Y$ .

Therefore, $∠ 1 ≅ ∠ 2$ and $∠ 3 ≅ ∠ 4$ , that is, $m ∠ 1 = m ∠ 2 and m ∠ 3 = m ∠ 4$ .

If the two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

It is given that, $\bar{X Y} ≅ \bar{X Z}$ therefore by isosceles triangle theorem $∠ X Y Z ≅ ∠ X Z Y$ .

But $m ∠ X Y Z = m ∠ 1 + m ∠ 2$ and $m ∠ X Z Y = m ∠ 3 + m ∠ 4$ .

Then $m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$ .

Substitute $m ∠ 2$ for $m ∠ 1$ and $m ∠ 3$ for $m ∠ 4$ into $m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$ and simplify.

$\begin{matrix} m ∠ 2 + m ∠ 2 = m ∠ 3 + m ∠ 3 \\ 2 m ∠ 2 = 2 m ∠ 3 \\ m ∠ 2 = m ∠ 3 \end{matrix}$

As $∠ 2 ≅ ∠ 3$ ,therefore by converse isosceles triangle theorem, $\bar{Y O} ≅ \bar{Z O}$ .

Hence it is proved that $\bar{Y O} ≅ \bar{Z O}$ .

Write a two-column proof based on above explanation.

Statement	Reason
$\bar{X Y} ≅ \bar{X Z}$	Given
$m ∠ 1 = m ∠ 2$	bisects
$m ∠ 3 = m ∠ 4$	bisects
$m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$	Isosceles triangle theorem
$m ∠ 2 = m ∠ 3$	Since
$\bar{Y O} ≅ \bar{Z O}$	Converse isosceles triangle theorem