Question

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figure poqr
$overline{or} \perp \overline{pq}$, $overline{ps} \cong \overline{qs}$

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figure fjhw
$overline{jw} \perp \overline{fh}$, $overline{fy} \cong \overline{hy}$, $overline{jy} \cong \overline{wy}$

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figure klmn
$overline{kn} \cong \overline{lm}$

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figure stuv
$overline{ts} \cong \overline{vu}$

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figure abcd
$overline{dc} \parallel \overline{ba}$, $overline{ad} \parallel \overline{cb}$

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figure ziab
$overline{zi} \cong \overline{ia} \cong \overline{ab} \cong \overline{bz}$

Explanation:

Since the problem involves analyzing geometric figures (like perpendicularity, congruent segments, parallel sides) and likely identifying properties or proving relationships, the subfield is Geometry (under Mathematics). Let's take Figure ABCD as an example to analyze:

Step1: Identify the figure type

Given \(\overline{DC} \parallel \overline{BA}\) and \(\overline{AD} \parallel \overline{CB}\), a quadrilateral with both pairs of opposite sides parallel is a parallelogram. So ABCD is a parallelogram.

Step2: Recall parallelogram properties

In a parallelogram, opposite sides are congruent (\(\overline{DC} \cong \overline{BA}\), \(\overline{AD} \cong \overline{CB}\)), opposite angles are congruent, and diagonals bisect each other.

(If another figure, e.g., Figure POQR: \(\overline{OR} \perp \overline{PQ}\) (right angle) and \(\overline{PS} \cong \overline{QS}\) (S is midpoint), so OR is the perpendicular bisector of PQ, implying \(\overline{OP} \cong \overline{OQ}\) and \(\overline{RP} \cong \overline{RQ}\) by perpendicular bisector theorem.)

Answer:

The subfield is Geometry (Mathematics), and for figure analysis (e.g., ABCD is a parallelogram, POQR has OR as perpendicular bisector, etc.).