Question

7.4

Answer 1

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. $\overline{QS} \cong \overline{RT}$ 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

$KM = \mathbf{\dfrac{3}{2} \cdot x+ 1}$

Which gives;

$\frac{3}{2} \cdot x+ 1 = 2\cdot x - 6$

$2\cdot x -\frac{3}{2} \cdot x = 6 + 1 = 7$

<em />$\dfrac{1}{2} \cdot x = 7$

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;

• WXYZ is a square

10. The two column proof is presented as follows;

Statement ${}$                                            Reason

PSUR is a rectangle ${}$                           Given

$\overline{PR} = \overline{SU}$ ${}$                                             Opposite sides of a rectangle

$\overline{PQ} \cong \overline{TU}$  ${}$                                           Given

$\overline{PQ} = \overline{TU}$  ${}$                                           Definition of congruency

$\overline{PR} = \overline{RQ} + \overline{PQ}$ ${}$                                  Segment addition postulate

$\overline{SU} = \overline{ST} + \overline{TU}$ ${}$                                   Segment addition postulate

$\overline{RQ} = \overline{ST}$    ${}$                                          Addition property of equality

ΔPQS ≅ ΔTUR    ${}$                                ASA rule of congruency

$\overline{QS} \cong \overline{RT}$         ${}$                                    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)

Learn more about quadrilaterals here:

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