Chapter 4, Problem 18CUR

To determine

Whether the given information be used to prove that two lines are parallel.

Answer to Problem 18CUR

  $\Delta ABC$ is isosceles by definition of isosceles triangle.

Explanation of Solution

Given information:

A paragraph proof: If $\overline{AX}$ is both a median and an altitude of $\Delta ABC$ , then $\Delta ABC$ is isosceles.

Calculation:

The two triangles are said to be congruent if they are copies of each other and if their vertices are superposed, then say that corresponding angles and the sides of the triangles are congruent.

Congruency of triangles can be proved by SSS, SAS and ASA, AAS and HL postulates.

To prove that two segments or two angles are congruent, firstly identity the two triangles in which that two segments or angles are corresponding parts. After that, prove that the identified triangles are congruent and then, state that the two parts are congruent using the reason “Corresponding parts of $\cong$ triangles are $\cong$ ”.

Firstly draw a rough figure by the information given in the problem.

  

As, $\overline{AX}$ is the median and altitude of $\Delta ABC$ , So $\overline{BX}=\overline{XC}$ & $m\angle AXB=m\angle AXC=90$ . This implies; $\overline{BX}=\overline{XC}$ & $\angle AXB=\angle AXC$ by definition of congruent segments and angles. Also $\overline{AX}\cong \overline{AX}$ by reflexive property and thus, $\Delta AXB\cong \Delta AXC$ by SAS postulate. Therefore, $\overline{AB}$ and $\overline{AC}$ are corresponding parts of congruent triangles. Thus, $\Delta ABC$ is isosceles by definition of isosceles triangle.