Question

∠uyv ≅ ∠vyw. complete the proof that overleftrightarrow{uw} ⊥ overleftrightarrow{vx}.

statement
1 ∠uyv ≅ ∠vyw
2 m∠uyv + m∠vyw = 180°
3 m∠uyv + m∠uyv = 180°
4 m∠uyv = 90°
5 overleftrightarrow{uw} ⊥ overleftrightarrow{vx}

reason

Explanation:

Step1: Given

Given that \( \angle UYV\cong\angle VYW\).

Step2: Definition of linear pair

Since \( \angle UYV\) and \( \angle VYW\) form a linear pair, \( m\angle UYV + m\angle VYW=180^{\circ}\) (by the linear - pair postulate).

Step3: Substitution property

Because \( \angle UYV\cong\angle VYW\), then \( m\angle UYV = m\angle VYW\). Substituting \( m\angle VYW\) with \( m\angle UYV\) in \( m\angle UYV + m\angle VYW = 180^{\circ}\), we get \( m\angle UYV+m\angle UYV = 180^{\circ}\) (substitution property).

Step4: Simplify the equation

Combining like terms in \( m\angle UYV+m\angle UYV = 180^{\circ}\), we have \( 2m\angle UYV=180^{\circ}\). Dividing both sides by 2, \( m\angle UYV = 90^{\circ}\) (division property of equality).

Step5: Definition of perpendicular lines

If the measure of the angle between two lines is \( 90^{\circ}\), then the lines are perpendicular. So, \(\overleftrightarrow{UW}\perp\overleftrightarrow{VX}\) (by the definition of perpendicular lines).

Answer:

1. Given; 2. Linear - pair postulate; 3. Substitution property; 4. Division property of equality; 5. Definition of perpendicular lines.