Question

if necessary, you may learn what the markings on a figure indicate.
(first figure: triangle abc with angle at a 42°, side ab and ac marked with ticks; triangle tuv with angle at t 42°, side tu and tv marked with ticks)
bc select uv
cannot be determined
(second figure: triangle deg with de=8, dg=7; triangle efg with ef=9, fg=8)
m∠deg select m∠fge
cannot be determined

Explanation:

Step1: Analyze the first triangle pair

In $\triangle ABC$ and $\triangle TUV$:

• $AC = TV$ (marked with two ticks), $AB = TU$ (marked with one tick), $\angle A=\angle T = 42^\circ$.
• By SAS (Side - Angle - Side) congruence criterion, $\triangle ABC\cong\triangle TUV$.
• Corresponding sides of congruent triangles are equal, so $BC = UV$.

Step2: Analyze the second triangle pair

In $\triangle DEG$ and $\triangle FGE$:

• $DE = FG = 8$, $EG$ is common. But $DG = 7$ and $EF = 9$.
• We have two sides equal ($DE = FG$, $EG = EG$) but the included angles? Wait, no, the third sides $DG = 7$ and $EF = 9$ are not equal. Wait, actually, we can use the Hinge Theorem (SAS Inequality Theorem). The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is shorter than the third side of the second triangle, then the included angle of the first triangle is smaller than the included angle of the second triangle.
• In $\triangle DEG$ and $\triangle FGE$: $DE = FG = 8$, $EG = EG$ (common side), and $DG = 7$, $EF = 9$. Since $DG

Answer:

First comparison: $BC = UV$; Second comparison: $m\angle DEG