Chapter 4.2, Problem 17WE

To determine

To prove: $\Delta RSV\cong \Delta UTV$

Explanation of Solution

Given information: $\overline{RS}\perp \overline{ST},\overline{TU}\perp \overline{ST}$ , and V is the midpoint of $\overline{ST}$ .

Proof:

  

Statement 1:

  $\overline{RS}\perp \overline{ST},\overline{TU}\perp \overline{ST}$

Reason 1:

Given

Statement 2: $m\angle S=90°,m\underset{\_}{}=90°$

Reason 2:

As $\overline{RS}\perp \overline{ST},\overline{TU}\perp \overline{ST}$ ,

  $m\angle S=90°,m\underset{\_}{\angle T}=90°$ (two lines are perpendicular if they form a right angle)

Statement 3: $\angle S\cong \angle T$

Reason 3:

As $m\angle S=90°,m\angle T=90°$ , $m\angle S=m\angle T$

So,

  $\angle S\cong \angle T$ (equal angles are also congruent)

Statement 4:

V is the midpoint of $\overline{ST}$

Reason 4:

Given

Statement 5:

  $\overline{SV}\cong \underset{\_}{}$

Reason 5:

As V is the midpoint of $\overline{ST}$ and midpoint is a point on a line segment that divides it into two congruent line segments.

So,

  $\overline{SV}\cong \underset{\_}{\overline{VT}}$

Statement 6: $\angle RVS\cong \underset{\_}{}$

Reason 6:

  $\overline{ST}$ and $\overline{RU}$ intersect at V.

So,

  $\angle RVS\cong \underset{\_}{\angle UVT}$ (vertically opposite angles are equal)

Statement 7: $\Delta \underset{\_}{}\cong \Delta \underset{\_}{}$

Reason 7:

  $\angle S\cong \angle T$ (from statement 3.)

  $\overline{SV}\cong \overline{VT}$ (from statement 5.)

  $\angle RVS\cong \angle UVT$ (from statement 6.)

So, $\Delta RSV\cong \Delta UTV$ (ASA congruence rule)

(according to ASA congruence rule, two triangles are said to be congruent if two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle.)

Statements	Reasons
1. $\overline{RS}\perp \overline{ST},\overline{TU}\perp \overline{ST}$ 2. $m\angle S=90°,m\underset{\_}{\angle T}=90°$ 3. $\angle S\cong \angle T$ 4. V is the midpoint of $\overline{ST}$ 5. $\overline{SV}\cong \underset{\_}{\overline{VT}}$ 6. $\angle RVS\cong \underset{\_}{\angle UVT}$ 7. $\Delta \underset{\_}{RSV}\cong \Delta \underset{\_}{UTV}$	1. Given 2. $\overline{RS}\perp \overline{ST},\overline{TU}\perp \overline{ST}$ 3. $m\angle S=m\angle T=90°$ 4. Given 5. V is the midpoint of $\overline{ST}$ 6. Vertically opposite angles 7. ASA congruence rule