b Use the Pythagorean Theorem and consider the congruent triangles from part A.
A
a
Statements
Reasons
1.
QR ⊥ QT, ST ⊥ QT, QT∥ SR, △ VSR is isosceles with base SR, and QT∥ SR
1.
Given
2.
∠ RQV and ∠ STV are right angles
2.
Definition of perpendicular lines
3.
∠ RQV ≅ ∠ STV
3.
All the right angles are congruent
4.
VR ≅ VS
4.
Definition of isosceles
5.
∠ VSR ≅ ∠ VRS
5.
Isosceles Triangle Theorem
6.
∠ QVR ≅ ∠ VRS and ∠ TVS ≅ ∠ VSR
6.
Alternate Interior Angle Theorem
7.
∠ QVR ≅ ∠ TVS
7.
Transitive Property of Congruence
8.
△ RQV ≅ △ STV
8.
Angle-Angle-Side Theorem
B
b 3 meters
Practice makes perfect
a A portion of the roller coaster track shown below.
We will prove that △ RQV ≅ △ STV. Therefore, we will construct a two column proof. In our first step, we will state the given statements and the statement that we will prove.
Given: QR ⊥ QT, ST ⊥ QT, QT∥ SR, and △ VSR is isosceles with base SR, and QT∥ SR
Prove: △ RQV ≅ △ STV
Using the given statements QR ⊥ QT and ST ⊥ QT, we can write the second step of the proof.
2. Definition of perpendicular lines
∠ RQV and ∠ STV are right angles
Since all the right angles are congruent, let's write the third step.
3. All the right angles are congruent
∠ RQV ≅ ∠ STV
We know that △ VSR is isosceles with base SR. Therefore, its legs will be congruent.
4. Definition of isosceles
VR ≅ VS
Next, we can use Isosceles Triangle Theorem to identify the congruent angles of △ VSR.
5. Isosceles Triangle Theorem
∠ VSR ≅ ∠ VRS
Because QT≅ RS, the sixth step of the proof can be written by the Alternate Interior Angle Theorem.
6. Alternate Interior Angle Theorem
∠ QVR ≅ ∠ VRS and ∠ TVS ≅ ∠ VSR
By the Transitive Property of Congruence and fourth step, we can write the seventh step.
7. Transitive Property of Congruence
∠ QVR ≅ ∠ TVS
As a result, two angles and the nonincluded side of △ RQV are congruent to the corresponding two angles and side of △ RQV. Thus, we can complete our proof by Angle-Angle-Side Theorem.
8. Angle-Angle-Side Theorem
△ RQV ≅ △ STV
Using these steps, let's construct the two column proof.
Statements
Reasons
1.
QR ⊥ QT, ST ⊥ QT, QT∥ SR, △ VSR is isosceles with base SR, and QT∥ SR
1.
Given
2.
∠ RQV and ∠ STV are right angles
2.
Definition of perpendicular lines
3.
∠ RQV ≅ ∠ STV
3.
All the right angles are congruent
4.
VR ≅ VS
4.
Definition of isosceles
5.
∠ VSR ≅ ∠ VRS
5.
Isosceles Triangle Theorem
6.
∠ QVR ≅ ∠ VRS and ∠ TVS ≅ ∠ VSR
6.
Alternate Interior Angle Theorem
7.
∠ QVR ≅ ∠ TVS
7.
Transitive Property of Congruence
8.
△ RQV ≅ △ STV
8.
Angle-Angle-Side Theorem
b We have been given that VR=2.5 meters and QR=2 meters.
To find the distance between QR and ST which is QT, we will use the Pythagorean Theorem in △ RQV to find QV.
Because congruent parts of congruent triangles are congruent, we can state that VT=1.5 meters. Now we can use the Segment Addition Postulate.
QV+VT=QT
Using this equation, let's find QT.
