c What is the sum of the interior angles of a quadrilateral?
D
d If the diagonals bisect each other they cut each other in half.
A
a False, see solution.
B
b True, see solution.
C
c True, see solution.
D
d False, see solution.
Practice makes perfect
a A parallelogram is a quadrilateral with two pairs of parallel sides. From the exercise we know that the quadrilateral has one pair of parallel sides. If the second pair of sides is congruent, the quadrilateral could be either a parallelogram or an isosceles trapezoid.
Therefore, the statement is false.
b In any polygon the sum of its interior angles can be calculated with the formula 180^(∘)(n-2). A quadrilateral has 4 sides, so if n=4 results in a product of 360^(∘) we know that the polygon must be a quadrilateral.
The polygon must be a quadrilateral. The statement is true.
c A rectangle has four right angles. From the exercise we know that 3 angles are right. Also, since the sum of the interior angles of the quadrilateral is 360^(∘), we can write and solve an equation containing the measure of the unknown angle.
3(90^(∘))+m∠ θ = 360^(∘) ⇔ m∠ θ = 90^(∘)
Since the last angle is also right, the quadrilateral is a rectangle.
d If the diagonals bisect each other, they cut each other in two equal halves. Let's first show that the diagonals of a rhombus actually do bisect each other. A rhombus is a quadrilateral with two pairs of parallel sides and four congruent sides. Let's draw such a figure including the diagonals.
As we can see, in a rhombus the diagonals do bisect each other. However, what happens if we enlarge one pair of sides forming a parallelogram? Let's try that.
As we can see, the diagonals still bisect each other when the quadrilateral is a parallelogram. Therefore, the statement is false.
