Question

given that \\(\overline{rt} \cong \overline{wx}\\), which statement must be true?\
options: \\(rt + tw = rx\\), \\(rt + tw = wx + tw\\), \\(rt = 2(rx)\\), \\(wx = 2(rx)\\)

Explanation:

Step1: Analyze Segment Addition

For a line segment \( RX \), if \( T \) is a point between \( R \) and \( X \), then \( RT + TW = RX \) (wait, no—correct segment addition: if \( T \) is between \( R \) and \( W \), and \( W \) is between \( R \) and \( X \), then \( RT + TW = RW \), but maybe the diagram has \( R - T - W - X \)? Wait, the key is \( \overline{RT} \cong \overline{WX} \) (so \( RT = WX \)). Now check each option:

• Option 1: \( RT + TW = RX \) – If \( T \) and \( W \) are on \( RX \) with \( R - T - W - X \), then \( RT + TW + WX = RX \), but \( RT = WX \), so \( RT + TW + RT = RX \), not \( RT + TW = RX \). Wait, maybe misinterpretation. Wait, the first option's label: "RT + TW = RX" – maybe the segment is \( R - T - W - X \), so \( RX = RT + TW + WX \), but \( RT = WX \), so \( RX = RT + TW + RT = 2RT + TW \), no. Wait, maybe the correct approach is to use the segment addition postulate and the congruent segments.

• Option 2: \( RT + TW = WX + TW \) – Subtract \( TW \) from both sides: \( RT = WX \), which is given by \( \overline{RT} \cong \overline{WX} \). So this must be true (since congruent segments have equal length, so \( RT = WX \), then adding \( TW \) to both sides preserves equality: \( RT + TW = WX + TW \)).

• Option 3: \( WX = 2(RX) \) – Since \( WX \) is part of \( RX \) (if \( W \) is between \( R \) and \( X \)), \( WX < RX \), so \( WX = 2RX \) is false.

• Option 4: \( RT = 2(RX) \) – \( RT \) is part of \( RX \), so \( RT < RX \), so \( RT = 2RX \) is false.

Wait, the second option (the one with "RT + TW = WX + TW") – let's re-express: \( RT + TW = WX + TW \) simplifies to \( RT = WX \), which is given by \( \overline{RT} \cong \overline{WX} \). So this is an application of the addition property of equality (adding \( TW \) to both sides of \( RT = WX \)).

Step2: Verify Each Option

• Option 1: \( RT + TW = RX \) – False, as \( RX \) would be longer (includes \( WX \) too, since \( RT = WX \)).
• Option 2: \( RT + TW = WX + TW \) – True, because \( RT = WX \) (congruent segments), so adding \( TW \) to both sides gives this equality.
• Option 3: \( WX = 2(RX) \) – False, \( WX \) is a part of \( RX \), so \( WX < RX \).
• Option 4: \( RT = 2(RX) \) – False, \( RT \) is a part of \( RX \), so \( RT < RX \).

So the correct option is the one with "RT + TW = WX + TW" (the third box? Wait, the labels: the four options are (top to bottom? Wait the image: top right: "RT + TW = RX", middle right: "RT + TW = WX + TW", top left: "WX = 2(RX)", bottom right: "RT = 2(RX)". So the middle right option: "RT + TW = WX + TW" is correct.

Answer:

The middle - right option (labeled "RT + TW = WX + TW")