Question

answer the following
a. construct quadrilaterals having exactly one, two, or four right angles.
b. can a quadrilateral have exactly three right angles? why?
c. can a parallelogram have exactly two right angles? why?
a. construct quadrilaterals having exactly one, two, or four right angles. which of the following shapes is a quadrilateral with exactly one right angle?

Explanation:

Step1: Recall quadrilateral angle - sum property

The sum of the interior angles of a quadrilateral is $360^{\circ}$.

Step2: Analyze part a

A right - angled trapezoid can have exactly one right angle. A trapezoid with two non - parallel sides and one right - angled corner is an example. A rectangle has four right angles. A trapezoid with two adjacent right angles has two right angles.

Step3: Analyze part b

Let the angles of the quadrilateral be $A$, $B$, $C$, and $D$. If $A = B=C = 90^{\circ}$, then $A + B+C=270^{\circ}$. Since $A + B + C+D = 360^{\circ}$, then $D=360^{\circ}-(A + B + C)=360^{\circ}-270^{\circ}=90^{\circ}$. So a quadrilateral cannot have exactly three right angles.

Step4: Analyze part c

In a parallelogram, opposite angles are equal. If a parallelogram has two right angles, then its opposite angles are also right angles, so it has four right angles (it is a rectangle or a square). So a non - rectangular parallelogram cannot have exactly two right angles.

Answer:

a. A right - angled trapezoid can have exactly one right angle. A trapezoid with two adjacent right angles has two right angles. A rectangle has four right angles.
b. No. Because if three angles are right angles, the fourth must also be a right angle due to the sum of interior angles of a quadrilateral being $360^{\circ}$.
c. No. In a parallelogram, opposite angles are equal. If two angles are right angles, all four are right angles.