Answer:
(5) ∠ FEH ≅ ∠ GHE : CPCTC
(6) ∠ FEH and ∠ GHE are supplementary : Consecutive angles of a parallelogram are supplementary.
(7) m∠ FEH = 90° : Congruent supplementary angles are right angles.
(8) EFGH is a rectangle : Definition of a rectangle.
Step-by-step explanation:
Given:
Quadrilateral EFGH is a parallelogram.
[tex]\overline{EG}\cong \overline{HF}[/tex]
Statements Reasons
1. EFGH is a parallelogram. Given
[tex]\overline{EG}\cong \overline{HF}[/tex]
2. [tex]\overline{EF}\cong \overline{GH}[/tex] If a quadrilateral is a parallelogram, then the opposite sides are congruent
3. [tex]\overline{EH}\cong \overline{EH}[/tex] Reflexive property of Congruence.
4. Δ EFH ≅ Δ HGE SSS Triangle Congruence Postulate.
Now, when two triangles are congruent by SSS, their corresponding angles are also congruent by CPCTC.
(5) ∠ FEH ≅ ∠ GHE CPCTC
Also, for a parallelogram, the same side angles sum is 180 degrees. ∠ FEH and ∠ GHE are supplementary as their sum is 180°.
(6) ∠ FEH and ∠ GHE are supplementary Consecutive angles of a parallelogram are supplementary.
Now, from statement (5), ∠ FEH ≅ ∠ GHE, so the supplementary pair are congruent. Therefore, each angle is equal to 90°.
Let ∠ FEH = ∠ GHE = [tex]x[/tex]. Then,
[tex]x+x=180\\2x=180\\x=\frac{180}{2}=90\°[/tex]
Therefore, ∠ FEH = ∠ GHE = 90°
(7) m∠ FEH = 90° Congruent supplementary angles are right angles.
Now, for a parallelogram with congruent diagonals, if any two consecutive angles are 90 degree each, then the remaining angles are also 90 degrees.
Now, if a parallelogram has all its angles equal to 90°, then the parallelogram is a rectangle from the definition of a rectangle.
(8) EFGH is a rectangle Definition of a rectangle.
