Question

tu ≅ vw, ∠quw ≅ ∠svt, and ∠t ≅ ∠w. complete the proof that st ≅ qw.

statement
1 tu ≅ vw
2 ∠quw ≅ ∠svt
3 ∠t ≅ ∠w
4 uw = vw + uv
5 tv = tu + uv
6 uw = tu + uv
7 tv = uw
8 △quw ≅ △svt
9 st ≅ qw

reason
given
given
given

additive property of length
substitution
transitive property of equality

cpctc

Explanation:

Step1: Determine the reason for statement 4

The formula \(UW = VW + UV\) is based on the Additive Property of Length. This property states that if a line segment \(UW\) is composed of two adjacent segments \(VW\) and \(UV\), then the length of \(UW\) is the sum of the lengths of \(VW\) and \(UV\).

Step2: Determine the reason for statement 8

We have \(TV = UW\) (from step 7), \(\angle T\cong\angle W\) (given as statement 3), and \(\angle QUW\cong\angle SVT\) (given as statement 2). By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle QUW\cong\triangle SVT\). In \(\triangle QUW\) and \(\triangle SVT\), \(\angle QUW\) and \(\angle SVT\) are the included angles between sides \(UW\) and \(TV\) (which are equal) and angles \(\angle W\) and \(\angle T\) (which are equal).

Answer:

1. Additive Property of Length; 8. ASA (Angle - Side - Angle)