Question

$overleftrightarrow{qr} perp overleftrightarrow{rs}$ and $overleftrightarrow{uv} perp overleftrightarrow{tu}$. complete the proof that $angle tuv cong angle qrs$.

	statement	reason
2	$overleftrightarrow{uv} perp overleftrightarrow{tu}$	given
3	$mangle qrs = 90^circ$	definition of perpendicular lines
4	$mangle tuv = 90^circ$
5	$mangle qrs = mangle tuv$
6	$angle tuv cong angle qrs$	definition of congruence

Explanation:

Step1: Determine Reason for Step 4

To find the reason for \( m\angle TUV = 90^\circ \), we use the definition of perpendicular lines, just as we did for \( m\angle QRS = 90^\circ \). Since \( \overleftrightarrow{UV} \perp \overleftrightarrow{TU} \), by the definition of perpendicular lines, the measure of the angle formed ( \( \angle TUV \)) is \( 90^\circ \).

Step2: Determine Reason for Step 5

For \( m\angle QRS = m\angle TUV \), we note that both angles are \( 90^\circ \) (from steps 3 and 4). So this is the Transitive Property of Equality (or simply substitution, since both equal \( 90^\circ \)). But more precisely, since \( m\angle QRS = 90^\circ \) and \( m\angle TUV = 90^\circ \), we can say \( m\angle QRS = m\angle TUV \) because they are both equal to the same value ( \( 90^\circ \) ), so the reason is "Substitution (or Transitive Property, as both are \( 90^\circ \))" or more accurately, "If two angles have the same measure (both \( 90^\circ \)), then their measures are equal". But following the pattern, since step 3 used the definition of perpendicular lines for \( \angle QRS \), step 4 will use the definition of perpendicular lines for \( \angle TUV \), and step 5 will be "Substitution (or Transitive Property)".

But let's fill in the blanks properly:

For step 4: Reason is "Definition of perpendicular lines" (because \( \overleftrightarrow{UV} \perp \overleftrightarrow{TU} \), so the angle between them is \( 90^\circ \)).

For step 5: Reason is "Substitution (or Transitive Property of Equality)" (since \( m\angle QRS = 90^\circ \) and \( m\angle TUV = 90^\circ \), so \( m\angle QRS = m\angle TUV \)).

Answer:

• Step 4 Reason: Definition of perpendicular lines
• Step 5 Reason: Substitution (or Transitive Property of Equality, as \( m\angle QRS = 90^\circ \) and \( m\angle TUV = 90^\circ \))