which triangle is △ abc similar to and why? note: figures are not drawn to scale.
△ abc is similar to △ def by aa similarity postulate.
△ abc is similar to △ ghi by aa similarity postulate.
△ abc is similar to △ jkl by aa similarity postulate.
△ abc is not similar to any of the triangles given.
(there are also triangle diagrams: △ abc with ∠a=20°, ∠b=80°; △ def with ∠d=18°, ∠e=80°; △ ghi with ∠g=20°, ∠h=70°; △ jkl with ∠k=80°, ∠l=80°)

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Accepted Answer

# Brief Explanations:
1. First, find the missing angle in $	riangle ABC$. The sum of angles in a triangle is $180^\circ$. In $	riangle ABC$, $\angle A = 20^\circ$, $\angle B = 80^\circ$, so $\angle C=180 - 20 - 80=80^\circ$.
2. For $	riangle DEF$: $\angle D = 18^\circ$, $\angle E = 80^\circ$, so $\angle F=180 - 18 - 80 = 82^\circ$. Angles don't match $	riangle ABC$.
3. For $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=180 - 20 - 70 = 90^\circ$. Angles don't match $	riangle ABC$. Wait, no, recalculate: Wait, $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=180 - 20 - 70 = 90^\circ$? No, wait, let's check again. Wait, in $	riangle ABC$, angles are $20^\circ$, $80^\circ$, $80^\circ$. For $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=180 - 20 - 70 = 90^\circ$? No, that's wrong. Wait, no, the third option: Wait, the triangles: Wait, $	riangle JKL$ has two $80^\circ$ angles? Wait, no, the options: Wait, the second option: $	riangle ABC$ and $	riangle GHI$: Wait, $\angle A = 20^\circ$, $\angle G = 20^\circ$. $\angle B = 80^\circ$, what about $	riangle GHI$'s angles? Wait, $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=90^\circ$? No, that's not matching. Wait, no, I made a mistake. Wait, in $	riangle ABC$, angles are $\angle A = 20^\circ$, $\angle B = 80^\circ$, so $\angle C=80^\circ$ (since $180 - 20 - 80 = 80$). Now, for $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=180 - 20 - 70 = 90^\circ$? No, that's not. Wait, the second option: Wait, maybe I misread the angles. Wait, the triangle $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=90^\circ$? No. Wait, the first option: $	riangle DEF$: $\angle D = 18^\circ$, $\angle E = 80^\circ$, so $\angle F=82^\circ$. Not matching. Wait, the third option: $	riangle JKL$: $\angle K = 80^\circ$, $\angle L = 80^\circ$, so $\angle J=20^\circ$ (since $180 - 80 - 80 = 20$). Oh! Wait, I missed $	riangle JKL$. Wait, the options: The third option is $	riangle ABC$ similar to $	riangle JKL$? Wait, no, the options are:

- $	riangle ABC$ similar to $	riangle DEF$: $	riangle DEF$ has $\angle D = 18^\circ$, $\angle E = 80^\circ$, so $\angle F=82^\circ$. Not matching $	riangle ABC$'s angles ($20^\circ$, $80^\circ$, $80^\circ$).

- $	riangle ABC$ similar to $	riangle GHI$: $	riangle GHI$ has $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I=90^\circ$. Not matching.

- $	riangle ABC$ similar to $	riangle JKL$: $	riangle JKL$ has $\angle K = 80^\circ$, $\angle L = 80^\circ$, so $\angle J=20^\circ$ (since $180 - 80 - 80 = 20$). So $	riangle JKL$ has angles $20^\circ$, $80^\circ$, $80^\circ$, same as $	riangle ABC$. So by AA similarity (two angles equal: $\angle A = \angle J = 20^\circ$, $\angle B = \angle K = 80^\circ$), so they are similar. Wait, but the option says $	riangle JKL$? Wait, no, the options: Wait, the third option is $	riangle ABC$ similar to $	riangle JKL$ by AA? Wait, the original options: Let me re - check the options:

The options are:

1. $	riangle ABC$ similar to $	riangle DEF$ by AA.

2. $	riangle ABC$ similar to $	riangle GHI$ by AA.

3. $	riangle ABC$ similar to $	riangle JKL$ by AA.

4. Not similar.

Wait, I think I misread the triangles. Let's re - examine:

- $	riangle ABC$: $\angle A = 20^\circ$, $\angle B = 80^\circ$, so $\angle C = 80^\circ$ (sum of angles in triangle is $180^\circ$: $180 - 20 - 80 = 80$).

- $	riangle DEF$: $\angle D = 18^\circ$, $\angle E = 80^\circ$, so $\angle F = 180 - 18 - 80 = 82^\circ$. Angles: $18^\circ$, $80^\circ$, $82^\circ$. Not matching $	riangle ABC$'s angles.

- $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I = 180 - 20 - 70 = 90^\circ$. Angles: $20^\circ$, $70^\circ$, $90^\circ$. Not matching.

- $	riangle JKL$: $\angle K = 80^\circ$, $\angle L = 80^\circ$, so $\angle J = 180 - 80 - 80 = 20^\circ$. Angles: $20^\circ$, $80^\circ$, $80^\circ$. Which matches $	riangle ABC$'s angles. So by AA similarity (two angles: $\angle A=\angle J = 20^\circ$, $\angle B=\angle K = 80^\circ$), so $	riangle ABC$ is similar to $	riangle JKL$ by AA. Wait, but the option says $	riangle JKL$? Wait, the original options: Wait, the third option is " $	riangle ABC$ is similar to $	riangle JKL$ by AA Similarity Postulate". So that's the correct one. Wait, but earlier I thought $	riangle GHI$, but no. Wait, maybe I made a mistake in the angle of $	riangle GHI$. Wait, $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$, so $\angle I = 90^\circ$. Not matching. $	riangle JKL$: $\angle J = 20^\circ$, $\angle K = 80^\circ$, $\angle L = 80^\circ$. So same as $	riangle ABC$. So the correct option is the third one? Wait, no, the options: Wait, the user's options: Let me check again.

Wait, the options are:

- $	riangle ABC$ is similar to $	riangle DEF$ by AA.

- $	riangle ABC$ is similar to $	riangle GHI$ by AA.

- $	riangle ABC$ is similar to $	riangle JKL$ by AA.

- Not similar.

Wait, $	riangle JKL$ has two $80^\circ$ angles and one $20^\circ$ angle, same as $	riangle ABC$ (which has $20^\circ$, $80^\circ$, $80^\circ$). So by AA (two angles: $20^\circ$ and $80^\circ$), they are similar. So the correct option is the third one: $	riangle ABC$ is similar to $	riangle JKL$ by AA Similarity Postulate. Wait, but wait, the triangle $	riangle JKL$: in the diagram, $\angle K = 80^\circ$, $\angle L = 80^\circ$, so $\angle J = 20^\circ$. So $	riangle ABC$ (angles $20^\circ$, $80^\circ$, $80^\circ$) and $	riangle JKL$ (angles $20^\circ$, $80^\circ$, $80^\circ$) have two angles equal, so AA similarity applies.

Wait, but earlier I thought $	riangle GHI$, but no. So the correct answer is the third option: $	riangle ABC$ is similar to $	riangle JKL$ by AA Similarity Postulate. Wait, but the options: Wait, the user's options: Let me check the original problem again.

Wait, the triangles:

- $	riangle ABC$: $\angle A = 20^\circ$, $\angle B = 80^\circ$.

- $	riangle DEF$: $\angle D = 18^\circ$, $\angle E = 80^\circ$.

- $	riangle GHI$: $\angle G = 20^\circ$, $\angle H = 70^\circ$.

- $	riangle JKL$: $\angle K = 80^\circ$, $\angle L = 80^\circ$.

So $	riangle JKL$ has $\angle J = 20^\circ$ (since $180 - 80 - 80 = 20$), so $\angle J = 20^\circ$, $\angle K = 80^\circ$, which matches $\angle A = 20^\circ$ and $\angle B = 80^\circ$ in $	riangle ABC$. So by AA (two angles), they are similar. So the correct option is the third one: $	riangle ABC$ is similar to $	riangle JKL$ by AA Similarity Postulate.

# Answer:
$	riangle ABC$ is similar to $	riangle JKL$ by AA Similarity Postulate.

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