Question

1. if one - pair of opposite sides of a quadrilateral are both parallel and congruent, what can you conclude?

a. the quadrilateral is a rectangle
b. the quadrilateral is a parallelogram
c. the quadrilateral is a trapezoid
d. the quadrilateral is a rhombus

1. which of the following is not a step in constructing an equilateral triangle inscribed in a circle?

a. measure the angles of the triangle with a protractor
b. mark off three equal arcs along the circles circumference
c. connect the points where the arcs intersect the circle
d. draw the circle using a compass

1. what is the property of the arcs that are drawn from two points on the angles sides when constructing an angle bisector?

a. they must be equal in length
b. they must be parallel
c. they must be perpendicular
d. they must intersect

1. in a parallelogram, consecutive angles add up to ______ degrees.

a. 90
b. 360
c. 270
d. 180

1. use the following points (2,3) and (6,7) to find the mid - point between them.

a. (-4,5)
b. (5,4)
c. (6, - 8)
d. (4,5)

1. when constructing a square inscribed in a circle, how are the four vertices located?

a. by drawing four arcs from a single point
b. by dividing the circle into four unequal parts
c. by creating the center and marking four equal sections
d. by constructing perpendicular bisectors from each diameter point

1. according to the perpendicular bisector conjecture, in triangle cdef, if segment dg is the perpendicular bisector of side ef and ef measures 14 units, what are the measures of segment eg and segment gf?

a. eg = 8 units, gf = 6 units
b. eg = 6 units, gf = 8 units
c. eg = 7 units, gf = 7 units
d. eg = 7 units, gf = 7 units

Explanation:

Step1: Recall quadrilateral properties

If one - pair of opposite sides of a quadrilateral are both parallel and congruent, by definition, the quadrilateral is a parallelogram. A rectangle has all angles as right - angles (not given here), a trapezoid has only one pair of parallel sides (not both parallel and congruent), and a rhombus has all sides congruent (not given here). So for question 1, the answer is b.

Step2: Analyze steps of constructing an equilateral triangle inscribed in a circle

To construct an equilateral triangle inscribed in a circle, we first draw the circle with a compass, mark off three equal arcs along the circle's circumference, and connect the points where the arcs intersect the circle. Measuring the angles with a protractor is not a step in the construction process. So for question 2, the answer is a.

Step3: Understand angle - bisector construction

When constructing an angle bisector, the arcs drawn from two points on the angle's sides must be equal in length so that we can find the point of intersection which helps in drawing the angle bisector. So for question 3, the answer is a.

Step4: Recall parallelogram angle properties

In a parallelogram, consecutive angles are supplementary, which means they add up to 180 degrees. So for question 4, the answer is d.

Step5: Use mid - point formula

The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For points \((2,3)\) and \((6,7)\), \(x=\frac{2 + 6}{2}=4\) and \(y=\frac{3+7}{2}=5\). The mid - point is \((4,5)\) (none of the given options are correct as there seems to be a typo in the options).

Step6: Consider square - in - circle construction

When constructing a square inscribed in a circle, we first draw a diameter of the circle, then construct a perpendicular bisector of the diameter. The four intersection points of the diameter, its perpendicular bisector, and the circle are the vertices of the square. This is equivalent to bisecting the circle into four equal parts. So for question 6, the answer is b.

Answer:

1. B. The quadrilateral is a parallelogram
2. A. Measure the angles of the triangle with a protractor
3. A. They must be equal in length
4. D. 180
5. (There is an error in the options provided as the correct mid - point of \((2,3)\) and \((6,7)\) is \((4,5)\))
6. B. By bisecting the circle into four equal parts