contestada

7.4
Practice A
In Exercises 1-5, the diagonals of rhombus ABCD intersect at E. Given that
mZEAD - 67', CE - 5, and DE 12, find the indicated measure.
1. MZAED
5
2. MZADE
E
12
3. MZBAE
67
44E
5. BE
In Exercises 6 and 7, find the lengths of the diagonals of rectangle JKLM.
6. JL = 3r + 4
7. JL = 2x - 6
KM = 4x - 1
KM
*+1
In Exercises 8 and 9, decide whether quadrilateral WXYZ is a rectangle, a
rhombus, or a square. Give all names that apply. Explain your reasoning.
8. W(3, 1), X(3,-2), Y(-5, -2), 2(-5, 1) 9. W(4, 1), X (1,4), Y(-2, 1), Z(1, -2)
Р
o
R
a
10. Use the figure to write a two-column proof.
Given: PSUR is a rectangle.
PQ E TU
Prove: OS = RT
S
U
11. In the figure, all sides are congruent and all angles are right angles.
B
a. Determine whether the quadrilateral is a rectangle.
Explain your reasoning.
b. Determine whether the quadrilateral is a rhombus.
Explain your reasoning
c. Determine whether the quadrilateral is a square.
Explain your reasoning.
d. Find m2AEB.
D
e. Find mZEAD.

74 Practice A In Exercises 15 the diagonals of rhombus ABCD intersect at E Given that mZEAD 67 CE 5 and DE 12 find the indicated measure 1 MZAED 5 2 MZADE E 12 class=

Respuesta :

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;

  • WXYZ is a square

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)

Learn more about quadrilaterals here:

https://brainly.com/question/12635945

Otras preguntas