Chapter 4.6, Problem 3WE

To determine

To prove: $\angle G\cong \angle T$ by writing reasons for the given key steps

Explanation of Solution

Given information: Steps of proof are given as follows:

 a. $\Delta RAJ\cong \Delta NAK$

 b. $\overline{RJ}\cong \overline{NK}$

 c. $\Delta GRJ\cong \Delta TNK$

 d. $\angle G\cong \angle T$

Proof:

  

Step a. $\Delta RAJ\cong \Delta NAK$

Consider $\Delta RAJ,\Delta NAK$

  $\overline{RA}\cong \overline{AN}$ (Given)

  $m\angle RJA=m\angle NKA=90°$ ( $\overline{RJ}\perp \overline{AG},\overline{NK}\perp \overline{AT}$ )

  $\angle RJA\cong \angle NKA$ (right angles are congruent to each other)

  $\angle RAJ\cong \angle NAK$ (vertically opposite angles)

So,

  $\Delta RAJ\cong \Delta NAK$ (by AAS congruence criteria)

(According to AAS congruence criteria, two triangles are said to be congruent if two angles and side not included between them one triangle are congruent to the two angles and side not included between them another triangle)

Step b. $\overline{RJ}\cong \overline{NK}$

  $\overline{RJ}\cong \overline{NK}$ (Corresponding parts of congruent triangles are equal)

Step c. $\Delta GRJ\cong \Delta TNK$

Consider $\Delta GRJ,\Delta TNK$

  $\overline{RJ}\cong \overline{NK}$ (from part b.)

  $m\angle RJG=m\angle NKT=90°$ ( $\overline{RJ}\perp \overline{AG},\overline{NK}\perp \overline{AT}$ )

  $\angle RJG\cong \angle NKT$ (right angles are congruent to each other)

  $\overline{GJ}\cong \overline{TK}$ (Given)

So,

  $\Delta GRJ\cong \Delta TNK$ (SAS congruence criteria)

(According to SAS congruence criteria, two triangles are said to be congruent if two sides and angle included between them of one triangle are congruent to two sides and angle included between them of another triangle)

Step d. $\angle G\cong \angle T$

  $\angle G\cong \angle T$ (Corresponding parts of congruent triangles are equal)