Question

$\angle ptq \cong \angle qtr$. complete the proof that $\overleftrightarrow{pr} \perp \overleftrightarrow{qs}$.
diagram: horizontal line with points s, t, q (s---t---q), vertical line through t with p (up) and r (down) (p↑ t ↓r)
table with columns statement and reason:

1. $\angle ptq \cong \angle qtr$
2. $m\angle ptq + m\angle qtr = 180^\circ$
3. $m\angle ptq + m\angle ptq = 180^\circ$
4. $m\angle ptq = 90^\circ$
5. $\overleftrightarrow{pr} \perp \overleftrightarrow{qs}$

reason columns are empty

Explanation:

Step1: Identify Given Information

The problem states that \( \angle PTQ \cong \angle QTR \), so this is the given information.

Step2: Linear Pair Postulate

\( \angle PTQ \) and \( \angle QTR \) form a linear pair (they are adjacent and their non - common sides form a straight line \( PR \)), so by the Linear Pair Postulate, \( m\angle PTQ + m\angle QTR=180^{\circ} \).

Step3: Substitution Property

Since \( \angle PTQ \cong \angle QTR \), their measures are equal, i.e., \( m\angle PTQ = m\angle QTR \). Substitute \( m\angle QTR \) with \( m\angle PTQ \) in the equation \( m\angle PTQ + m\angle QTR = 180^{\circ} \), we get \( m\angle PTQ+m\angle PTQ = 180^{\circ} \) (which is \( 2m\angle PTQ=180^{\circ} \)).

Step4: Solve for \( m\angle PTQ \)

Divide both sides of the equation \( 2m\angle PTQ = 180^{\circ} \) by 2. So \( m\angle PTQ=\frac{180^{\circ}}{2}=90^{\circ} \) (using the Division Property of Equality).

Step5: Definition of Perpendicular Lines

If the measure of an angle formed by two lines (or rays) is \( 90^{\circ} \), then the lines (or rays) are perpendicular. Since \( m\angle PTQ = 90^{\circ} \), by the definition of perpendicular lines, \( \overleftrightarrow{PR}\perp\overleftrightarrow{QS} \).

Now, filling in the reasons:

1. Reason for \( \angle PTQ\cong\angle QTR \): Given
2. Reason for \( m\angle PTQ + m\angle QTR = 180^{\circ} \): Linear Pair Postulate (angles that form a linear pair are supplementary)
3. Reason for \( m\angle PTQ + m\angle PTQ = 180^{\circ} \): Substitution Property (substituting \( m\angle QTR \) with \( m\angle PTQ \) because \( \angle PTQ\cong\angle QTR \))
4. Reason for \( m\angle PTQ = 90^{\circ} \): Solving \( 2m\angle PTQ = 180^{\circ} \) (Division Property of Equality, divide both sides by 2)
5. Reason for \( \overleftrightarrow{PR}\perp\overleftrightarrow{QS} \): Definition of Perpendicular Lines (if two lines intersect to form a \( 90^{\circ} \) angle, they are perpendicular)

Answer:

1. Given
2. Linear Pair Postulate
3. Substitution Property
4. Division Property of Equality (solving \( 2m\angle PTQ = 180^{\circ} \))
5. Definition of Perpendicular Lines