Question

→
tw bisects ∠uwy and ∠vtx. complete the proof that △tvw ≅ △txw.

statement
1 →
tw bisects ∠uwy
2 →
tw bisects ∠vtx
3 ∠xwy ≅ ∠uwv
4 ∠twy ≅ ∠twu
5 ∠wtx ≅ ∠vtw
6 m∠twx = m∠twy + m∠xwy
7 m∠twv = m∠twu + m∠uwv
8 m∠twx = m∠twu + m∠uwv
9 m∠twv = m∠twx
10 →
tw ≅ →
tw
11 △tvw ≅ △txw

reason
given
given
vertical angle theorem

definition of angle bisector
additive property of angle measure

substitution
transitive property of equality
reflexive property of congruence
asa

Explanation:

Step1: Analyze Statement 4

We know that \(\overrightarrow{TW}\) bisects \(\angle UWY\) (from Statement 1). By the definition of an angle bisector, an angle bisector divides an angle into two congruent angles. So, \(\angle TWY\) and \(\angle TWU\) are the two angles formed by \(\overrightarrow{TW}\) bisecting \(\angle UWY\), hence \(\angle TWY\cong\angle TWU\) by the definition of angle bisector.

Step2: Analyze Statement 7

The Additive Property of Angle Measure states that if a point lies in the interior of an angle, the measure of the angle is the sum of the measures of the two smaller angles formed. For \(\angle TVW\), the point \(U\) (or \(W\) related to the angles) lies in the interior such that \(\angle TVW\) is composed of \(\angle TWU\) and \(\angle UWV\). So, \(m\angle TVW = m\angle TWU + m\angle UWV\) by the Additive Property of Angle Measure.

Answer:

For Statement 4, the reason is "Definition of angle bisector". For Statement 7, the reason is "Additive Property of Angle Measure".