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?
b. can a quadrilateral have exactly three right angles? why?
a. no. if the quadrilateral has three right angles, then the fourth must also be a right angle because the sum of the measures of the four angles is 360°
b. no. if the quadrilateral has three right angles, then the fourth must also be a right angle because the sum of the measures of the four angles is 360°
c. yes. if the quadrilateral has three right angles, then the fourth must be 110° because the sum of the measures of the four angles is 360°
d. yes. if the quadrilateral has three right angles, then the fourth must be 110° because the sum of the measures of the four angles is 360°

Explanation:

Step1: Recall angle - sum property of quadrilaterals

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

Step2: Analyze the case of three right - angles in a quadrilateral

Let the three right - angles each measure $90^{\circ}$. The sum of these three angles is $3\times90^{\circ}=270^{\circ}$. Let the fourth angle be $x$. Then, by the angle - sum property of quadrilaterals, $270^{\circ}+x = 360^{\circ}$, so $x=360^{\circ}- 270^{\circ}=90^{\circ}$. So a quadrilateral cannot have exactly three right - angles.

Step3: Analyze the case of a parallelogram with two right - angles

In a parallelogram, opposite angles are equal. If a parallelogram has two right - angles, then its opposite angles are also right - angles, and it becomes a rectangle. So a parallelogram cannot have exactly two non - opposite right - angles. If the two right - angles are opposite, then all angles are right - angles and it is a rectangle.

Answer:

a. For exactly one right - angle: Draw a non - rectangular quadrilateral with one $90^{\circ}$ angle. For exactly two right - angles: Draw a trapezoid with two adjacent right - angles. For exactly four right - angles: Draw a rectangle or a square.
b. A. No. If the quadrilateral has three right - angles, then the fourth must also be a right - angle because the sum of the measures of the four angles is $360^{\circ}$.
c. No. In a parallelogram, opposite angles are equal. If two angles are right - angles, then all four angles are right - angles (since opposite angles are equal), and it is a rectangle, not a non - rectangular parallelogram.