Question

use figure abcd to answer the questions below.

1. which side is opposite ab?
2. name two sides that are adjacent to bc.
3. name the two diagonals.
4. which angle is opposite ∠abc?
5. the measure of ∠dab is 80°. the measure of ∠abc is 114°. the remaining two angles are equal in measure. what is the measure of ∠adc?

use the properties of quadrilaterals to answer each question.

1. name three properties that rectangles and squares have in common.
2. true or false a square, a rectangle, and a rhombus are three examples of parallelograms. explain your answer.
3. a four - sided figure has equal and parallel opposite sides. none of the four angles are right angles, but all four sides are equal in length. what is the name of the figure?
4. true or false a trapezoid can never have a right angle. explain your answer.

imagine you are viewing the quadrilateral to the right from an angle that distorts the figure. you know that wx and yz are equal, xy and wz are equal, xw and wz are equal, and ∠xwz is a right angle. use the information to answer the questions below.

1. is it possible for this figure to have only one right angle? why or why not?
2. what is the name for this figure?
3. explore tiles are often made in the shape of squares because it is easy to fit squares together to cover a surface. would it be possible to use rhombus - shaped tiles to cover a surface? how about trapezoid tiles?

Explanation:

Step1: Recall angle - sum property of quadrilaterals

The sum of the interior angles of a quadrilateral is 360°.

Step2: Set up an equation

Let the measure of ∠ADC and ∠BCD be \(x\) (since they are equal). We know that ∠DAB = 80° and ∠ABC = 114°. So, \(80 + 114+x + x=360\).

Step3: Simplify the equation

Combining like - terms, we get \(194 + 2x=360\).

Step4: Solve for \(x\)

Subtract 194 from both sides: \(2x=360 - 194=166\). Then divide both sides by 2: \(x = 83\).

Step5: Answer question 6

Rectangles and squares both have four right angles, opposite sides are equal and parallel, and the diagonals are equal.

Step6: Answer question 7

True. A square has all sides equal, opposite sides parallel, and all angles 90°. A rectangle has opposite sides equal and parallel and all angles 90°. A rhombus has all sides equal and opposite sides parallel. By definition, a parallelogram has opposite sides parallel. So, a square, rectangle, and rhombus are parallelograms.

Step7: Answer question 8

The figure is a rhombus. A rhombus has equal and parallel opposite sides and all sides are equal in length, and it does not have to have right angles.

Step8: Answer question 9

False. A trapezoid can have a right angle. A right - trapezoid has two right angles.

Step9: Answer question 10

No. In a quadrilateral, if one angle is 90° and \(WX = YZ\), \(XY = WZ\), \(XW = WZ\), the figure is a square or a rectangle. In a rectangle or square, if one angle is 90°, all angles are 90° due to the properties of parallel sides and the angle - sum property of quadrilaterals.

Step10: Answer question 11

The figure is a square. Since \(WX = YZ\), \(XY = WZ\), \(XW = WZ\) and ∠XWZ = 90°, it satisfies the properties of a square (all sides equal and one right angle implies all angles are right angles because of parallel sides).

Step11: Answer question 12

Yes, it is possible to use rhombus - shaped tiles to cover a surface. The sum of the interior angles of a rhombus is 360°, and by arranging them properly, we can fill the plane. Yes, it is also possible to use trapezoid tiles to cover a surface. We can arrange trapezoids in a pattern such that the angles at the vertices add up to 360° and cover the surface.

Answer:

1. DC
2. AB and CD
3. AC and BD
4. ∠ADC
5. 83°
6. Four right angles; Opposite sides are equal and parallel; Diagonals are equal
7. True. They all have opposite sides parallel.
8. Rhombus
9. False. A right - trapezoid has right angles.
10. No. One right angle and equal - side conditions imply a square/rectangle with all right angles.
11. Square
12. Yes for rhombus - shaped tiles. Yes for trapezoid tiles.