Question

complete the proof that \overleftrightarrow{qs} \perp \overleftrightarrow{pr}.

(image of intersecting lines qs and pr at point t, with angle ptq and qtr marked)

	statement	reason
1	$\angle qtr \cong \angle ptq$	given
2	$m\angle ptq + m\angle qtr = 180^\circ$
3	$m\angle ptq + m\angle ptq = 180^\circ$
4	$m\angle ptq = 90^\circ$	properties of addition, subtraction, multiplication, and division
5	$\overleftrightarrow{qs} \perp \overleftrightarrow{pr}$

Explanation:

Step1: Reason for Statement 2

Angles on a straight line are supplementary, so their measures sum to \(180^\circ\). Here, \(\angle PTQ\) and \(\angle QTR\) form a linear pair (they are adjacent and their non - common sides form a straight line \(\overleftrightarrow{PR}\)), so \(m\angle PTQ + m\angle QTR=180^\circ\) (Linear Pair Postulate).

Step2: Reason for Statement 3

Since \(\angle QTR\cong\angle PTQ\) (given), their measures are equal, i.e., \(m\angle QTR = m\angle PTQ\). By substituting \(m\angle QTR\) with \(m\angle PTQ\) in the equation \(m\angle PTQ + m\angle QTR = 180^\circ\) (from Step 2), we get \(m\angle PTQ + m\angle PTQ=180^\circ\) (Substitution Property).

Step3: Reason for Statement 4

The equation \(m\angle PTQ + m\angle PTQ = 180^\circ\) can be rewritten as \(2m\angle PTQ=180^\circ\). Using the Division Property of Equality (dividing both sides by 2), we get \(m\angle PTQ = 90^\circ\).

Step4: Reason for Statement 5

If two lines intersect to form a right angle (an angle with measure \(90^\circ\)), then the lines are perpendicular. Since \(m\angle PTQ = 90^\circ\), \(\overleftrightarrow{QS}\) and \(\overleftrightarrow{PR}\) intersect at \(T\) and form a right angle at \(T\), so \(\overleftrightarrow{QS}\perp\overleftrightarrow{PR}\) (Definition of Perpendicular Lines).

Answer:

• For Statement 2: Linear Pair Postulate
• For Statement 3: Substitution Property
• For Statement 4: Division Property of Equality
• For Statement 5: Definition of Perpendicular Lines