The properties of the quadrilaterals gives the equivalent relations used
to find the lengths and the angles.
Response:
1. m∠AED = 90°
2. m∠ADE = 23°
3. m∠BAE = 67°
4. AE = 5
5. BE = 12
6. 19
7. 22
8. WXYZ is a rectangle
9. WXYZ is a square
10. [tex]\overline{QS} \cong \overline{RT}[/tex] by Corresponding Parts of Congruent Triangles Congruent, CPCTC
11. a. The diagonals are perpendicular and the given figure is not a rectangle
b. The given figure is not a rhombus
c. The figure is a square.
d. ∠AEB = 90°
e. m∠EAD = 45°
Which properties of a quadrilateral can be used to find the required dimensions?
1. The diagonals of a rhombus bisect each other at right angles
Therefore;
m∠AED = 90° (by definition of right angles)
2. m∠EAD and m∠ADE are complementary angles
Which gives;
m∠EAD + m∠ADE = 90°
m∠ADE = 90° - m∠EAD
Therefore;
m∠ADE = 90° - 67° = 23°
3. The diagonals of a rhombus bisect the angles, therefore;
m∠BAE = m∠EAD = 67°
4. The diagonals bisect each other, therefore;
AE = CE = 5
5. BE = DE = 12
6. JL = 3·x + 4
KM = 4·x - 1
Which gives;
3·x + 4 = 4·x - 1
4·x - 3·x = 4 + 1 = 5
x = 5
KM = JL = 3 × 5 + 4 = 19
- The lengths of the diagonals of rectangle JKLM is 19
7. JL = 2·x - 6
[tex]KM = \mathbf{\dfrac{3}{2} \cdot x+ 1}[/tex]
Which gives;
[tex]\frac{3}{2} \cdot x+ 1 = 2\cdot x - 6[/tex]
[tex]2\cdot x -\frac{3}{2} \cdot x = 6 + 1 = 7[/tex]
[tex]\dfrac{1}{2} \cdot x = 7[/tex]
x = 2 × 7 = 14
JL = 2 × 14 - 6 = 22
- The lengths of the diagonals of rectangle JKLM are 22
8. W(3, 1), X(3, -2), Y(-5, -2), Z(-5, 1)
WX = 1 - (-2) = 3
YZ = 1 - (-2) = 3
XY = 3 - (-5) = 8
WZ = 3 - 5 = 8
Slope of WX = (1 - (-2)) ÷ (3 - 3) = ∞
Slope of YZ = (1 - (-2)) ÷ (-5 - (-5)) = ∞
Slope of XY = (-2- (-2)) ÷ (3- (-5)) = 0
Slope of WZ = (1- 1) ÷ (3- (-5)) = 0
Therefore;
WX and YZ are perpendicular to XY and WZ
The properties of WXYZ are the properties of a quadrilateral having perpendicular sides.
- Given that the sides are not equal, the figure is a rectangle
9. W(4, 1), X(1, 4), Y(-2, 1), Z(1, -2)
WX = √((4 - 1)² + (1 - 4)²) = 3·√2
YZ = √((1 - (-2))² + (-2 - 1)²) = 3·√2
XY = √((1 - (-2))² + (4 - 1)²) = 3·√2
WZ = √((1 - (-2))² + (4 - 1)²) = 3·√2
Slope of WX = (4 - 1) ÷ (1 - 4) = -1
Slope of YZ = (-2 - 1) ÷ (1 - (-2)) = -1
Slope of XY = (1 - 4) ÷ ((-2) - 1) = 1
Slope of WZ = (-2 - 1) ÷ (1 - 4) = 1
Therefore;
The lengths of the sides are equal and the sides are perpendicular to each other
Therefore;
10. The two column proof is presented as follows;
Statement [tex]{}[/tex] Reason
PSUR is a rectangle [tex]{}[/tex] Given
[tex]\overline{PR} = \overline{SU}[/tex] [tex]{}[/tex] Opposite sides of a rectangle
[tex]\overline{PQ} \cong \overline{TU}[/tex] [tex]{}[/tex] Given
[tex]\overline{PQ} = \overline{TU}[/tex] [tex]{}[/tex] Definition of congruency
[tex]\overline{PR} = \overline{RQ} + \overline{PQ}[/tex] [tex]{}[/tex] Segment addition postulate
[tex]\overline{SU} = \overline{ST} + \overline{TU}[/tex] [tex]{}[/tex] Segment addition postulate
[tex]\overline{RQ} = \overline{ST}[/tex] [tex]{}[/tex] Addition property of equality
ΔPQS ≅ ΔTUR [tex]{}[/tex] ASA rule of congruency
[tex]\overline{QS} \cong \overline{RT}[/tex] [tex]{}[/tex] CPCTC
11. The properties of a rectangle are;
The interior angles are 90°
The diagonals are not perpendicular to each other
In the given figure, the four triangles formed by the diagonals are congruent, therefore;
The angles at the vertex point of the four tringles are equal to each other and therefore, equal to 90°
Therefore;
- The diagonals are perpendicular and the given figure is not a rectangle
b. The lengths of the diagonals, are equal, therefore;
- The given figure is not a rhombus
e. The properties of the figure, which includes;
All sides are equal
The interior angles are 90°
- The diagonals are equal and bisect each other at 90° indicate that the figure is a square
d. The diagonals of a square are angle bisectors of the interior angles
Therefore;
m∠ABE = m∠CBE
m∠ABE + m∠CBE = 90° by definition of complementary angles
Therefore;
m∠ABE = 90° ÷ 2 = 45°
Similarly;
m∠ABE = m∠BAE = 45°
m∠AEB = 180° - (m∠ABE + m∠BAE)
- m∠AEB = 180° - (45° + 45°) = 90°
e. m∠EAD = m∠BAE = 45° (angles formed by the diagonal AC)
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