Question

what are the corresponding parts of the figures below? select all that apply. the figures are two triangles (or quadrilaterals? ) with labeled vertices c, d, e and p, q, o. there are checkboxes with options: ∠c ≅ ∠q, overline{cd} ≅ overline{pq}, overline{dc} ≅ overline{po}, overline{ce} ≅ overline{oq}

Explanation:

Step1: Analyze Angle Correspondence

Check the marked angles. $\angle C$ and $\angle O$ (wait, no, re - examine: The first triangle has $\angle C$, $\angle D$, $\angle E$; the second has $\angle P$, $\angle Q$, $\angle O$. Wait, the marked angles: $\angle D$ and $\angle P$ have the same marking, $\angle C$ and $\angle O$? Wait, no, the original check marks: Wait, the user's check marks might be wrong? Wait, no, let's re - do.

Wait, the first triangle: sides $CD$ (with two marks), $CE$ (two marks), $DE$ (one mark). The second triangle: $PQ$ (one mark), $OQ$ (two marks), $PO$ (two marks), $\angle P$ (marked), $\angle Q$ (marked), $\angle O$ (marked).

Wait, correct correspondence: $\angle D\cong\angle P$ (same marking), $\angle E\cong\angle Q$ (same marking), $\angle C\cong\angle O$? Wait, no, the initial check marks in the problem might be incorrect. Wait, no, let's look at the sides:

• Side $CD$: in triangle $CDE$, side $CD$ has two tick marks. Side $PO$ in triangle $POQ$ has two tick marks. So $\overline{DC}\cong\overline{PO}$ (since $DC$ is same as $CD$).
• Side $CE$: two tick marks, side $OQ$: two tick marks, so $\overline{CE}\cong\overline{OQ}$.
• Angle $\angle C$: in triangle $CDE$, angle at $C$; angle $\angle O$: in triangle $POQ$, angle at $O$? Wait, no, the angle at $Q$: $\angle Q$ is a right - like angle? Wait, maybe the correct correspondences are:

Wait, the correct corresponding parts:

1. $\overline{DC}\cong\overline{PO}$ (both have two tick marks on their sides)
2. $\overline{CE}\cong\overline{OQ}$ (both have two tick marks)
3. $\angle D\cong\angle P$ (same marking), but the option given is $\angle C\cong\angle Q$ which is wrong. Wait, maybe the original problem's check marks are incorrect, but according to the options:

Wait, the options are:

• $\angle C\cong\angle Q$: No, should be $\angle C\cong\angle O$ or $\angle D\cong\angle P$, $\angle E\cong\angle Q$.
• $\overline{CD}\cong\overline{PQ}$: $CD$ has two marks, $PQ$ has one mark, so no.
• $\overline{DC}\cong\overline{PO}$: $DC$ (same as $CD$) has two marks, $PO$ has two marks, so yes.
• $\overline{CE}\cong\overline{OQ}$: $CE$ has two marks, $OQ$ has two marks, so yes.

Wait, the initial check marks in the problem are wrong. But according to the correct analysis:

The correct corresponding parts are $\overline{DC}\cong\overline{PO}$, $\overline{CE}\cong\overline{OQ}$, and the angle $\angle D\cong\angle P$, $\angle E\cong\angle Q$, $\angle C\cong\angle O$. But among the given options:

• $\angle C\cong\angle Q$: Incorrect.
• $\overline{CD}\cong\overline{PQ}$: Incorrect (different number of marks).
• $\overline{DC}\cong\overline{PO}$: Correct (same number of marks).
• $\overline{CE}\cong\overline{OQ}$: Correct (same number of marks).

So the correct options are $\overline{DC}\cong\overline{PO}$ and $\overline{CE}\cong\overline{OQ}$, and the angle correspondence is wrong in the initial check. But if we go by the problem's options, the correct ones (after re - evaluation) are:

Step2: Re - evaluate Each Option

• Option $\angle C\cong\angle Q$: Incorrect. The angle at $C$ should correspond to angle at $O$, angle at $Q$ corresponds to angle at $E$.
• Option $\overline{CD}\cong\overline{PQ}$: Incorrect. $CD$ has two marks, $PQ$ has one mark.
• Option $\overline{DC}\cong\overline{PO}$: Correct. $DC$ (same as $CD$) has two marks, $PO$ has two marks.
• Option $\overline{CE}\cong\overline{OQ}$: Correct. $CE$ has two marks, $OQ$ has two marks.

Answer:

$\overline{DC}\cong\overline{PO}$, $\overline{CE}\cong\overline{OQ}$ (Note: The initial check for $\angle C\cong\angle Q$ and $\overline{CD}\cong\overline{PQ}$ is incorrect. The correct corresponding parts among the given options are $\overline{DC}\cong\overline{PO}$ and $\overline{CE}\cong\overline{OQ}$)