كويز تفاعلي: Geometry Assessment: Triangle Congruence and Angle Measures

The figure shows two pairs of congruent sides ($GL \cong JL$ and $HL \cong KL$) and the included angles are vertical angles, which are always congruent. Therefore, the triangles are congruent by SAS.

Only one pair of congruent sides is marked ($QR \cong TS$). There is no information provided about the other sides or the angles, so neither SSS nor SAS can be used.

The triangles share a side ($\overline{AC}$) and are given two pairs of congruent sides with congruent included angles, thus satisfying the SAS Congruence Postulate.

Although the triangles have congruent angles, no information about the side lengths or their congruency is provided, meaning SSS or SAS cannot be applied.

For $\triangle LKM$: $m\angle 2 = 180^\circ - (57^\circ + 71^\circ) = 52^\circ$. Angle 1 and $57^\circ$ form a linear pair, so $m\angle 1 = 180^\circ - 57^\circ = 123^\circ$. For $\triangle JLK$: $m\angle 3 = 180^\circ - (123^\circ + 28^\circ) = 29^\circ$.

In \right $\triangle RTW$: $m\angle 1 = 180^\circ - (90^\circ + 60^\circ) = 30^\circ$. In \right $\triangle RSW$: $m\angle 2 = 180^\circ - (90^\circ + 30^\circ) = 60^\circ$.

In $\triangle MPQ$: $m\angle 1 = 180^\circ - (66^\circ + 58^\circ) = 56^\circ$. $\angle 2$ is a vertical angle to $\angle 1$, so $m\angle 2 = 56^\circ$. In $\triangle NQO$: $m\angle 3 = 180^\circ - (56^\circ + 50^\circ) = 74^\circ$.

Based on the internal angles provided in the diagram, vertical angles and the triangle sum theorem are used. Specifically, if $m\angle 3 = 71^\circ$, then its vertical angle is also $71^\circ$. In the triangle with $80^\circ$, we find $m\angle 2 = 180^\circ - (80^\circ + 71^\circ) = 29^\circ$. Then $m\angle 1 = 180^\circ - 71^\circ = 109^\circ$ as they are supplementary.

Assuming the triangle is isosceles based on the diagram, the sum of angles is $180^\circ$. Therefore, the base angles are calculated as: $(180^\circ - 146^\circ) / 2 = 34^\circ / 2 = 17^\circ$.