Question

uv ⊥ vw and st ⊥ rs. complete the proof that ∠rst ≅ ∠uvw.

statement
1 uv ⊥ vw
2 st ⊥ rs
3 m∠uvw = 90°
4 m∠rst = 90°
5 m∠uvw = m∠rst
6 ∠rst ≅ ∠uvw

reason
given
given
definition of perpendicular lines

Explanation:

Step1: Identify the property for \(m\angle RST = 90^{\circ}\)

Since \(\overleftrightarrow{ST}\perp\overleftrightarrow{RS}\), by the definition of perpendicular lines (if two lines are perpendicular, the angle formed between them is \(90^{\circ}\)), we can fill the reason for statement 4.

Step2: Justify \(m\angle UVW=m\angle RST\)

From statements 3 (\(m\angle UVW = 90^{\circ}\)) and 4 (\(m\angle RST=90^{\circ}\)), by the substitution property of equality (if \(a = b\) and \(c = b\), then \(a = c\)), we get \(m\angle UVW=m\angle RST\).

Step3: Justify \(\angle RST\cong\angle UVW\)

From statement 5 (\(m\angle UVW=m\angle RST\)), by the definition of congruent angles (if the measures of two angles are equal, then the angles are congruent), we get \(\angle RST\cong\angle UVW\).

For statement 4, the reason is "Definition of perpendicular lines".
For statement 5, the reason is "Substitution property of equality".
For statement 6, the reason is "Definition of congruent angles".

Answer:

• Statement 4 Reason: Definition of perpendicular lines
• Statement 5 Reason: Substitution property of equality
• Statement 6 Reason: Definition of congruent angles