Question

which two triangles are congruent by the sss theorem? complete the congruence statement.
(image of three triangles: △dec, △ust, △ijh with congruency marks)
△□ ≅ △

Explanation:

Step1: Recall SSS Congruence

SSS (Side - Side - Side) congruence theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Step2: Analyze the Markings

- For $\triangle DEC$ (wait, actually the first triangle is $\triangle DEC$? Wait, no, the first triangle has vertices $D$, $E$, $C$. The markings: $DE$ has three marks, $DC$ has two marks, $EC$ has one mark.
- The second triangle is $\triangle UTS$: $UT$ has three marks, $US$? Wait, no, $UT$ has three marks, $TS$ has one mark, $US$ has three marks? Wait, no, looking at the second triangle: $UT$ (three marks), $TS$ (one mark), $US$ (three marks)? Wait, no, the third triangle is $\triangle IJH$: $IJ$ has two marks, $JH$ has one mark, $IH$ has three marks.
Wait, let's re - examine:
- $\triangle DEC$: $DE$ (three tick marks), $DC$ (two tick marks), $EC$ (one tick mark).
- $\triangle IJH$: $IJ$ (two tick marks), $JH$ (one tick mark), $IH$ (three tick marks).
- $\triangle UTS$: $UT$ (three tick marks), $TS$ (one tick mark), $US$ (three tick marks)? No, wait, $UT$ (three), $US$ (three), $TS$ (one). Wait, no, the first triangle: $DE$ (three), $DC$ (two), $EC$ (one). The third triangle: $IJ$ (two), $JH$ (one), $IH$ (three). The second triangle: $UT$ (three), $US$ (one), $TS$ (three). Wait, no, maybe I mislabeled. Wait, the first triangle: vertices $D$, $E$, $C$. Sides: $DE$ (three ticks), $DC$ (two ticks), $EC$ (one tick). The third triangle: vertices $I$, $J$, $H$. Sides: $IJ$ (two ticks), $JH$ (one tick), $IH$ (three ticks). The second triangle: vertices $U$, $T$, $S$. Sides: $UT$ (three ticks), $TS$ (one tick), $US$ (three ticks). Wait, no, the correct match: $\triangle DEC$ and $\triangle IJH$? Wait, no, wait $\triangle DEC$: $DE$ (three) - corresponds to $IH$ (three) in $\triangle IJH$? No, wait, let's check the number of tick marks:
- Three - tick sides: $DE$ (in $\triangle DEC$), $UT$ (in $\triangle UTS$), $IH$ (in $\triangle IJH$).
- Two - tick sides: $DC$ (in $\triangle DEC$), $IJ$ (in $\triangle IJH$).
- One - tick sides: $EC$ (in $\triangle DEC$), $JH$ (in $\triangle IJH$), $TS$ (in $\triangle UTS$).

Wait, $\triangle DEC$: sides with 3, 2, 1 ticks. $\triangle IJH$: sides with 3 (IH), 2 (IJ), 1 (JH) ticks. So by SSS, $\triangle DEC\cong\triangle IJH$? Wait, no, wait the first triangle is $\triangle DEC$? Wait, no, the first triangle is $\triangle D E C$ ( $D$, $E$, $C$), the third is $\triangle I J H$ ( $I$, $J$, $H$). Wait, or maybe the first triangle is $\triangle EDC$ and the third is $\triangle JIH$? Wait, let's do the correspondence:

- Side with 3 ticks: $DE$ (in $\triangle EDC$) and $IH$ (in $\triangle IJH$)
- Side with 2 ticks: $DC$ (in $\triangle EDC$) and $IJ$ (in $\triangle IJH$)
- Side with 1 tick: $EC$ (in $\triangle EDC$) and $JH$ (in $\triangle IJH$)

So the congruence is $\triangle EDC\cong\triangle IJH$? Wait, no, maybe the first triangle is $\triangle DEC$ and the third is $\triangle IJH$. Alternatively, maybe the first triangle is $\triangle DCE$ and the third is $\triangle IHJ$. Wait, perhaps the correct pair is $\triangle DEC$ and $\triangle IJH$. Wait, but let's check again.

Wait, the first triangle: $D$, $E$, $C$. Sides: $DE$ (three), $DC$ (two), $EC$ (one). The third triangle: $I$, $J$, $H$. Sides: $IH$ (three), $IJ$ (two), $JH$ (one). So by SSS, $\triangle DEC\cong\triangle IJH$ (or $\triangle EDC\cong\triangle JIH$ depending on vertex order). But maybe the first triangle is $\triangle DCE$ and the third is $\triangle IHJ$. Wait, perhaps the intended answer is $\triangle DEC\cong\triangle IJH$ or $\triangle EDC\cong\triangle JIH$. Wait, another way: the first triangle (with $D$, $E$, $C$) and the third triangle (with $I$, $J$, $H$) have sides with the same number of tick marks. So the congruence statement is $\triangle DEC\cong\triangle IJH$ (or $\triangle EDC\cong\triangle JIH$). But maybe the labels are $\triangle DCE$ and $\triangle IHJ$. Wait, perhaps the correct answer is $\triangle DEC\cong\triangle IJH$ (or more accurately, looking at the tick marks: the triangle with vertices $D$, $E$, $C$ and the triangle with vertices $I$, $J$, $H$ have corresponding sides equal. So the congruence is $\triangle DEC\cong\triangle IJH$ (or $\triangle EDC\cong\triangle JIH$). But maybe the first triangle is $\triangle DCE$ and the third is $\triangle IHJ$. Alternatively, maybe the first triangle is $\triangle EDC$ and the third is $\triangle JIH$.

Wait, maybe I made a mistake. Let's re - express:

- Triangle 1: $D$, $E$, $C$. Sides: $DE$ (3 ticks), $DC$ (2 ticks), $EC$ (1 tick).
- Triangle 3: $I$, $J$, $H$. Sides: $IH$ (3 ticks), $IJ$ (2 ticks), $JH$ (1 tick).

So the correspondence is $D
ightarrow I$, $E
ightarrow H$, $C
ightarrow J$? No, $DE$ (3) corresponds to $IH$ (3), $DC$ (2) corresponds to $IJ$ (2), $EC$ (1) corresponds to $JH$ (1). So the congruence is $\triangle DEC\cong\triangle IJH$ (since $DE = IH$, $DC = IJ$, $EC = JH$).

Answer:

$\triangle DEC\cong\triangle IJH$ (or $\triangle EDC\cong\triangle JIH$ depending on the order of vertices, but the most appropriate is $\triangle DEC\cong\triangle IJH$)