Question

$\angle twu \cong \angle tvx$, $\overline{tw} \cong \overline{tv}$, and $\angle utv \cong \angle wtx$. complete the proof that $\overline{uw} \cong \overline{vx}$.

statement	reason
2. $\overline{tw} \cong \overline{tv}$	given
3. $\angle utv \cong \angle wtx$	given
4. $m\angle vtx = m\angle vtw + m\angle wtx$
5. $m\angle utw = m\angle utv + m\angle vtw$
6. $m\angle vtx = m\angle vtw + m\angle utv$	substitution
7. $m\angle utw = m\angle vtx$	transitive property of equality
8. $\triangle tvx \cong \triangle twu$	asa
9. $\overline{uw} \cong \overline{vx}$	cpctc

Explanation:

Step1: Identify the Angle Addition Postulate

The Angle Addition Postulate states that if a point lies in the interior of an angle, then the sum of the measures of the two smaller angles formed is equal to the measure of the larger angle. In this case, for angle \( \angle VTX \), the point \( W \) is on \( XV \), so \( m\angle VTX = m\angle VTW + m\angle WTX \) by the Angle Addition Postulate.

Step2: For the fifth statement

Similarly, for angle \( \angle UTW \), the point \( V \) is on \( UW \), so \( m\angle UTW = m\angle UTV + m\angle VTW \) by the Angle Addition Postulate.

Answer:

For statement 4, the reason is "Angle Addition Postulate". For statement 5, the reason is "Angle Addition Postulate".