Triangle ABC has two angles measuring 59º and 88º. The third angle can be found by using the fact that the three angles of a triangle must sum up to 180º.
So, we have:
[tex]\begin{gathered} \angle A+\angle B+\angle C=180^{\circ} \\ \\ 88^{\circ}+\angle B+59^{\circ}=180^{\circ} \\ \\ 147^{\circ}+\angle B=180^{\circ} \\ \\ \angle B=180^{\circ}-147^{\circ} \\ \\ \angle B=33^{\circ} \end{gathered}[/tex]Now we know the measures of all angles and sides of the given triangle, let's analyze each of the congruence cases.
• For, triangle DEF, we see that:
Side DE is congruent to side AB (because they are represented using the same symbol)
Side EF is congruent to side BC (again, they both are represented by the same symbol)
The angle between sides DE and EF is congruent to the angle between sides AB and BC (they both measure 33º).
Thus, triangle DEF has a pair of sides, and the angle between them congruent two a pair of sides and the angle between them of triangle ABC.
Therefore, those two triangles are congruent by the Side-Angle-Side theorem. Or, for short: SAS.
• For ,triangle GHI ,we see that:
It has a pair of angles, 33º and 59º, that are congruent to a pair of angles of triangle ABC. Also, the side HI between those angles is congruent to side BC.
Thus, triangle GHI and triangle ABC are congruent by the Angle-Side-Angle Theorem. Or, for short: ASA.
For triangle
