Question

statement reason 1 ∠uwy ≅ ∠uxv given 2 ∠vuw ≅ ∠xuy given 3 overline{uw} ≅ overline{ux} given 4 m∠wuy = m∠wux + m∠xuy 5 m∠vux = m∠vuw + m∠wux additive property of angle measure 6 m∠wuy = m∠wux + m∠vuw substitution 7 m∠vux = m∠wuy transitive property of equality 8 △uwy ≅ △uxv 9 overline{wy} ≅ overline{vx}

Explanation:

Step1: Identify the property for step 4

The formula \(m\angle WUY=m\angle WUX + m\angle XUY\) is based on the Additive Property of Angle Measure. This property states that if an angle is composed of two non - overlapping adjacent angles, then the measure of the larger angle is the sum of the measures of the two smaller adjacent angles. In this case, \(\angle WUY\) is composed of \(\angle WUX\) and \(\angle XUY\).

Step2: Justify step 8

To prove \(\triangle UWY\cong\triangle UXV\), we use the ASA (Angle - Side - Angle) congruence criterion.
We know that \(\angle UWY\cong\angle UXV\) (given), \(\overline{UW}\cong\overline{UX}\) (given), and \(m\angle VUX = m\angle WUY\) (from step 7, which means \(\angle VUX\cong\angle WUY\)).
In \(\triangle UWY\) and \(\triangle UXV\), we have two angles and the included side equal.

Step3: Justify step 9

Since \(\triangle UWY\cong\triangle UXV\) (from step 8), by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) property, \(\overline{WY}\cong\overline{VX}\).

Answer:

• Step 4 Reason: Additive Property of Angle Measure
• Step 8 Reason: ASA (Angle - Side - Angle) Congruence Criterion
• Step 9 Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)