Question

yz ≅ wx, tz ≅ vw, tu ≅ uv, and ux ≅ uy. complete the proof that △txz ≅ △vyw.

1 (overline{yz} cong overline{wx}) given
2 (overline{tz} cong overline{vw}) given
3 (overline{tu} cong overline{uv}) given
4 (overline{ux} cong overline{uy}) given
5 (tx = tu + ux) additive property of length
6 (vy = uv + uy)
7 (tx = uv + uy) substitution
8 (tx = vy)
9 (xz = yz + xy) additive property of length
10 (wy = wx + xy) additive property of length
11 (xz = wx + xy) substitution
12 (wy = xz) transitive property of equality
13 (△txz cong △vyw)

Explanation:

Step1: Analyze the reason for \(VY = UV + UY\)

Since it is about the length of a line segment composed of two adjacent line segments, the reason is "Additive Property of Length".

Step2: Analyze the reason for \(TX = VY\)

We know \(TX=UV + UY\) (from step7) and \(VY = UV + UY\) (from step6). By substitution (if \(a=b\) and \(a = c\), then \(b = c\)), the reason is "Substitution".

Step3: Analyze the reason for \(\triangle TXZ\cong\triangle VYW\)

We have \(TX = VY\) (from step8), \(XZ=WY\) (from step12), and \(TZ = VW\) (from step2). According to the Side - Side - Side (SSS) congruence criterion for triangles (if three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent), the reason is "SSS Congruence".

Answer:

1. Additive Property of Length; 8. Substitution; 13. SSS Congruence