Question

$overline{rs} cong overline{st}$ and $overline{ru} cong overline{tu}$. complete the proof that $overline{su} perp overline{rt}$.

(image of a quadrilateral-like figure with vertices u, t, s, r and a point v inside)

	statement	reason
2	$overline{ru} cong overline{tu}$
3	$overline{su} cong overline{su}$
4	$\triangle rsu cong \triangle tsu$
5	$angle rsu cong angle tsu$
6	$overline{sv} cong overline{sv}$
7	$\triangle rsv cong \triangle tsv$
8	$angle rvs cong angle svt$
9	$mangle rvs + mangle svt = 180^circ$
10	$mangle rvs + mangle rvs = 180^circ$
11	$mangle rvs = 90^circ$
12	$overline{su} perp overline{rt}$

Explanation:

Step1: Identify Given Information

We are given \(\overline{RS} \cong \overline{ST}\) (Statement 1) and \(\overline{RU} \cong \overline{TU}\) (Statement 2). Also, \(\overline{SU}\) is a common side, so \(\overline{SU} \cong \overline{SU}\) (Statement 3, by Reflexive Property of Congruence).

Step2: Prove \(\triangle RSU \cong \triangle TSU\)

Using the SSS (Side - Side - Side) Congruence Criterion, since we have three pairs of congruent sides: \(\overline{RS} \cong \overline{ST}\), \(\overline{RU} \cong \overline{TU}\), and \(\overline{SU} \cong \overline{SU}\), we can conclude that \(\triangle RSU \cong \triangle TSU\) (Statement 4, Reason: SSS Congruence Criterion).

Step3: Corresponding Angles of Congruent Triangles

Since \(\triangle RSU \cong \triangle TSU\), their corresponding angles \(\angle RSU\) and \(\angle TSU\) are congruent (Statement 5, Reason: Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Step4: Identify Another Common Side

\(\overline{SV}\) is a common side to \(\triangle RSV\) and \(\triangle TSV\), so \(\overline{SV} \cong \overline{SV}\) (Statement 6, Reason: Reflexive Property of Congruence).

Step5: Prove \(\triangle RSV \cong \triangle TSV\)

We know that \(\overline{RS} \cong \overline{ST}\) (Statement 1), \(\angle RSU \cong \angle TSU\) (Statement 5), and \(\overline{SV} \cong \overline{SV}\) (Statement 6). By the SAS (Side - Angle - Side) Congruence Criterion, \(\triangle RSV \cong \triangle TSV\) (Statement 7, Reason: SAS Congruence Criterion).

Step6: Corresponding Angles of Newly Congruent Triangles

Since \(\triangle RSV \cong \triangle TSV\), their corresponding angles \(\angle RVS\) and \(\angle SVT\) are congruent (Statement 8, Reason: CPCTC).

Step7: Linear Pair of Angles

\(\angle RVS\) and \(\angle SVT\) form a linear pair, so \(m\angle RVS + m\angle SVT=180^{\circ}\) (Statement 9, Reason: Linear Pair Postulate).

Step8: Substitute Congruent Angles

Since \(\angle RVS \cong \angle SVT\), we can substitute \(m\angle SVT\) with \(m\angle RVS\) in the equation \(m\angle RVS + m\angle SVT = 180^{\circ}\). So \(m\angle RVS+m\angle RVS = 180^{\circ}\) (Statement 10, Reason: Substitution Property, since \(m\angle SVT = m\angle RVS\)).

Step9: Solve for \(m\angle RVS\)

Simplify the equation \(2m\angle RVS=180^{\circ}\). Dividing both sides by 2, we get \(m\angle RVS = 90^{\circ}\) (Statement 11, Reason: Division Property of Equality).

Step10: Definition of Perpendicular Lines

If the measure of an angle between two lines is \(90^{\circ}\), then the lines are perpendicular. Since \(m\angle RVS = 90^{\circ}\), \(\overline{SU}\perp\overline{RT}\) (Statement 12, Reason: Definition of Perpendicular Lines, because if two lines intersect to form a \(90^{\circ}\) angle, they are perpendicular).

Answer:

The completed proof with reasons is as follows:

Statement	Reason
2. \(\overline{RU} \cong \overline{TU}\)	Given
3. \(\overline{SU} \cong \overline{SU}\)	Reflexive Property of Congruence
4. \(\triangle RSU \cong \triangle TSU\)	SSS Congruence Criterion
5. \(\angle RSU \cong \angle TSU\)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
6. \(\overline{SV} \cong \overline{SV}\)	Reflexive Property of Congruence
7. \(\triangle RSV \cong \triangle TSV\)	SAS Congruence Criterion
8. \(\angle RVS \cong \angle SVT\)	CPCTC
9. \(m\angle RVS + m\angle SVT = 180^{\circ}\)	Linear Pair Postulate
10. \(m\angle RVS + m\angle RVS = 180^{\circ}\)	Substitution Property (since \(m\angle SVT=m\angle RVS\))
11. \(m\angle RVS = 90^{\circ}\)	Division Property of Equality (\(2m\angle RVS = 180^{\circ}\Rightarrow m\angle RVS=\frac{180^{\circ}}{2} = 90^{\circ}\))
12. \(\overline{SU}\perp\overline{RT}\)	Definition of Perpendicular Lines (if two lines intersect at a \(90^{\circ}\) angle, they are perpendicular)