Question

9 multiple choice 1 point
a quadrilateral must be a parallelogram if one pair of opposite sides is
parallel and the other pair of opposite sides is congruent
congruent and parallel
parallel, only
congruent, only
10 multiple choice 1 point
which statement is always true?
parallelograms are rectangles.
rhombiare squares.
rectangles are squares.
squares are rectangles.
11 multiple choice 1 point
the length of one side of a square is 13 feet. what is the length, to the nearest foot,
of a diagonal of the square?
18
26
13
19

Explanation:

Question 9

A quadrilateral is a parallelogram if one pair of opposite sides is both congruent and parallel. This is a fundamental property of parallelograms. If only parallel (without congruence) or only congruent (without parallelism) for one pair of opposite sides, it may not be a parallelogram. For example, a trapezoid has one pair of parallel sides but is not a parallelogram.

• A parallelogram is a rectangle only when its angles are right - angles. So, “Parallelograms are rectangles” is not always true.
• A rhombus is a square only when its angles are right - angles. So, “Rhombi are squares” is not always true.
• A rectangle is a square only when its adjacent sides are equal. So, “Rectangles are squares” is not always true.
• A square has all the properties of a rectangle (opposite sides equal, all angles right - angles). So, “Squares are rectangles” is always true.

Step1: Use the Pythagorean theorem

In a square of side length \(a = 13\) feet, the diagonal \(d\) forms a right - triangle with two sides of the square. By the Pythagorean theorem \(d^{2}=a^{2}+a^{2}\). Since \(a = 13\), we have \(d^{2}=13^{2}+13^{2}\).

$$d^{2}=169 + 169=338$$

Step2: Solve for \(d\)

Take the square root of both sides: \(d=\sqrt{338}\). We know that \(\sqrt{324}=18\) and \(\sqrt{361} = 19\). Since \(338\approx324+(338 - 324)=324 + 14\), and \(\sqrt{338}\approx18.4\)

Answer:

congruent and parallel

Question 10