57. decide which of the following statements about quadrilaterals are true. select all that apply.
□ a. a square is always a parallelogram.
□ b. a parallelogram is always a rectangle.
□ c. a rectangle is always a parallelogram.
□ d. a rhombus is always a square.
□ e. a square is always a rhombus.
□ f. a parallelogram is always a trapezoid.
□ g. a parallelogram is always a rhombus.
58. which of the following provides enough information to prove that the quadrilateral is a parallelogram? select all that apply.
(image of quadrilateral defg)
□ a. $\overline{de} \cong \overline{fg}, \overline{ef} \cong \overline{gd}$
□ b. $\overline{ef} \cong \overline{gd}, \overline{ef} \parallel \overline{gd}$
□ c. $\overline{de} \parallel \overline{fg}, \overline{ef} \parallel \overline{gd}$
□ d. $\overline{de} \parallel \overline{fg}, \overline{ef} \cong \overline{gd}$
□ e. $\overline{de} \cong \overline{ef}, \overline{gf} \cong \overline{dg}$
59. which of the following quadrilaterals must have perpendicular diagonals? select all that apply.
□ a. rectangle
□ b. rhombus
□ c. parallelogram
□ d. square
□ e. trapezoid
60. in the diagram, abcd is a parallelogram. which measures are correct? select all that apply.
(image of parallelogram abcd with angles and lengths labeled)
□ a. $cd = 16$
□ b. $eb = 7$
□ c. $bd = 17.2$
□ d. $eb = 10.2$
□ e. $m\angle abc = 60^\circ$
□ f. $m\angle bcd = 120^\circ$
□ g. $m\angle abd = 43^\circ$
□ h. $m\angle bdc = 17^\circ$

Question

Accepted Answer

### Question 57
# Explanation:
## Step 1: Recall Quadrilateral Definitions
- **Square**: 4 equal sides, 4 right angles, opposite sides parallel (so a parallelogram).
- **Parallelogram**: Opposite sides parallel and equal, opposite angles equal.
- **Rectangle**: 4 right angles, opposite sides parallel/equal (so a parallelogram).
- **Rhombus**: 4 equal sides, opposite sides parallel (so a parallelogram).
- **Trapezoid**: At least one pair of parallel sides.

## Step 2: Evaluate Each Option
- **A**: Square has parallel opposite sides (parallelogram definition). True.
- **B**: Parallelogram doesn’t need right angles (rectangle does). False.
- **C**: Rectangle has parallel opposite sides. True.
- **D**: Rhombus doesn’t need right angles (square does). False.
- **E**: Square has 4 equal sides (rhombus definition). True.
- **F**: Parallelogram has two pairs of parallel sides (trapezoid: at least one pair). True (by inclusive trapezoid definition).
- **G**: Parallelogram doesn’t need equal sides (rhombus does). False.

# Answer: A, C, E, F

### Question 58
# Explanation:
## Step 1: Recall Parallelogram Theorems
A quadrilateral is a parallelogram if:
- Both pairs of opposite sides are parallel ($ \parallel $) and/or equal ($ \cong $).

## Step 2: Evaluate Each Option
- **A**: $ \overline{DE} \cong \overline{FG}, \overline{EF} \cong \overline{GD} $ (both pairs of opposite sides equal). Sufficient.
- **B**: $ \overline{EF} \cong \overline{GD}, \overline{EF} \parallel \overline{GD} $ (one pair of sides equal and parallel). Sufficient.
- **C**: $ \overline{DE} \parallel \overline{FG}, \overline{EF} \parallel \overline{GD} $ (both pairs of opposite sides parallel). Sufficient.
- **D**: $ \overline{DE} \parallel \overline{FG}, \overline{EF} \cong \overline{GD} $ (one pair parallel, one pair equal). Sufficient (theorem: one pair parallel + equal ⇒ parallelogram).
- **E**: $ \overline{DE} \cong \overline{EF}, \overline{GF} \cong \overline{DG} $ (adjacent sides equal, not opposite). Insufficient.

# Answer: A, B, C, D

### Question 59
# Explanation:
## Step 1: Recall Diagonal Properties
- **Rectangle**: Diagonals equal, not necessarily perpendicular.
- **Rhombus**: Diagonals are perpendicular (definition: rhombus diagonals bisect at 90°).
- **Parallelogram**: Diagonals bisect each other, not necessarily perpendicular.
- **Square**: Special rhombus (and rectangle), so diagonals perpendicular.
- **Trapezoid**: Diagonals not necessarily perpendicular.

## Step 2: Evaluate Each Option
- **A**: Rectangle: diagonals not perpendicular. False.
- **B**: Rhombus: diagonals perpendicular. True.
- **C**: Parallelogram: diagonals not necessarily perpendicular. False.
- **D**: Square: is a rhombus, so diagonals perpendicular. True.
- **E**: Trapezoid: diagonals not necessarily perpendicular. False.

# Answer: B, D

### Question 60
# Explanation:
## Step 1: Recall Parallelogram Properties
- Opposite sides equal: $ AB = CD, AD = BC $.
- Diagonals bisect each other: $ AE = EC, DE = EB $.
- Adjacent angles supplementary: $ \angle DAB + \angle ABC = 180^\circ $, $ \angle DAB + \angle BCD = 180^\circ $ (wait, no: $ \angle DAB $ and $ \angle BCD $ are equal, $ \angle ABC $ and $ \angle ADC $ are equal).
- Triangle angle sum: $ 180^\circ $.

## Step 2: Evaluate Each Option
- **A**: $ CD = AB = 16 $? Wait, $ AB = 16 $, but $ AD = 7 $, so $ CD = AB = 16 $? Wait, no—wait, $ AB $ and $ CD $ are opposite sides. Wait, diagram shows $ AB = 16 $, $ BC = 7 $. So $ CD = AB = 16 $? Wait, no, $ AD = BC = 7 $, $ AB = CD = 16 $. Wait, but let’s check:
  - **A**: $ CD = AB = 16 $. True (opposite sides of parallelogram).
  - **B**: $ EB = DE $? Wait, diagonals bisect each other, so $ DE = EB $? Wait, diagram shows $ DE = 10.2 $? Wait, no—wait, diagram: $ DE = 10.2 $, $ EC = 7 $? Wait, no, $ E $ is the intersection of diagonals. Wait, $ DE = 10.2 $, so $ EB = DE = 10.2 $? Wait, no, maybe I misread. Wait, the diagram: $ DE = 10.2 $, $ EC = 7 $? No, $ BC = 7 $, $ AD = 7 $. Wait, diagonals bisect each other, so $ AE = EC $, $ DE = EB $. Wait, $ DE = 10.2 $, so $ EB = 10.2 $? But option B says $ EB = 7 $. False. Wait, maybe I messed up. Wait, let’s re-express:

Wait, the diagram: $ AB = 16 $, $ BC = 7 $, $ DE = 10.2 $, $ EC = 7 $? No, $ BC = 7 $, so $ AD = 7 $. Diagonals: $ AC $ and $ BD $ intersect at $ E $. So $ AE = EC $, $ DE = EB $.

- **A**: $ CD = AB = 16 $. True (opposite sides).
- **B**: $ EB = 7 $? If $ DE = 10.2 $, then $ EB = 10.2 $. False. Wait, maybe the diagram has $ DE = 10.2 $, $ EB = 7 $? No, that contradicts diagonals bisecting. Wait, maybe I misread. Let’s check other options:

- **C**: $ BD = DE + EB $. If $ DE = 10.2 $, $ EB = 10.2 $, then $ BD = 20.4 $. But option C says $ BD = 17.2 $. False. Wait, maybe the diagram is $ DE = 10.2 $, $ EC = 7 $, so $ AE = 7 $, $ DE = 10.2 $, so $ EB = 10.2 $. So **B** is false.

- **D**: $ EB = 10.2 $. True (diagonals bisect, so $ DE = EB = 10.2 $).
- **E**: $ \angle ABC $: $ \angle DAB = 120^\circ $, so $ \angle ABC = 180^\circ - 120^\circ = 60^\circ $. True.
- **F**: $ \angle BCD = \angle DAB = 120^\circ $ (opposite angles of parallelogram). True.
- **G**: $ \angle ABD $: In $ \triangle ABD $, $ \angle ADB = 43^\circ $, $ \angle DAB = 120^\circ $, so $ \angle ABD = 180 - 120 - 43 = 17^\circ $. So $ \angle ABD = 17^\circ $, not 43°. False.
- **H**: $ \angle BDC $: $ \angle ADC = \angle ABC = 60^\circ $ (opposite angles), $ \angle ADB = 43^\circ $, so $ \angle BDC = 60^\circ - 43^\circ = 17^\circ $. True.

Wait, let’s re-express:
- **A**: $ CD = AB = 16 $. True (opposite sides).
- **B**: $ EB = 7 $. False (diagonals bisect, $ DE = 10.2 $, so $ EB = 10.2 $).
- **C**: $ BD = 17.2 $. $ DE = 10.2 $, $ EB = 10.2 $, so $ BD = 20.4 $. False.
- **D**: $ EB = 10.2 $. True (diagonals bisect).
- **E**: $ \angle ABC = 60^\circ $. $ \angle DAB = 120^\circ $, adjacent angles supplementary: $ 180 - 120 = 60 $. True.
- **F**: $ \angle BCD = 120^\circ $. $ \angle DAB = 120^\circ $, opposite angles equal. True.
- **G**: $ \angle ABD = 43^\circ $. $ \angle ADB = 43^\circ $, but $ \angle ABD $ is not equal (triangle angle sum: $ 180 - 120 - 43 = 17 $). False.
- **H**: $ \angle BDC = 17^\circ $. $ \angle ADC = 60^\circ $, $ \angle ADB = 43^\circ $, so $ 60 - 43 = 17 $. True.

So correct options: A, D, E, F, H? Wait, no—wait, let’s recheck:

Wait, $ AB = 16 $, so $ CD = AB = 16 $ (A: true).

$ DE = 10.2 $, so $ EB = DE = 10.2 $ (D: true, B: false).

$ \angle ABC = 180 - 120 = 60^\circ $ (E: true).

$ \angle BCD = \angle DAB = 120^\circ $ (F: true).

$ \angle BDC = 60 - 43 = 17^\circ $ (H: true).

# Answer: A, D, E, F, H

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QUESTION IMAGE

Question

Explanation:

Question 57

Step 1: Recall Quadrilateral Definitions

Step 2: Evaluate Each Option

Step 1: Recall Parallelogram Theorems

Step 2: Evaluate Each Option

Step 1: Recall Diagonal Properties

Step 2: Evaluate Each Option

Answer:

Question 58