Quadrilateral ABCD is a parallelogram. If adjacent angles are congruent, which statement must be true?
A. Quadrilateral ABCD is a square.
B. Quadrilateral ABCD is a rhombus.
C. Quadrilateral ABCD is a rectangle.
D. Quadrilateral ABCD is an isosceles trapezoid.
My first pick is A but A is not the right answer. Correct answer is C. Could someone please explain to me I am confused. Thank you for your help.
a rectangle is a parallelogram with supplementary opposite angles (and congruent adjacent angles)
a square is a rectangle with equal sides
in a parallelogram, adjacent angles are supplementary.
If they are also congruent, they must be right angles.
So, C is correct. You know nothing about the side lengths.
Amanda is constructing equilateral triangle JKL inscribed in circle M. To construct the inscribed polygon, she is going to use a compass to partition the circle into congruent arcs. To what width should she set the compass when partitioning the circle?
A.
The width must be equal to the radius of circle M.
B.
The width must be equal the diameter of circle M.
C.
The width can be equal to either the radius or the diameter of circle M.
D.
The width can be any size greater than the radius but less than the diameter of circle M.
E.
The width must be less than the radius of circle M.
Well, well, well, looks like we have a geometry question here! Let's shed some light on this parallelogram puzzle.
If we have a parallelogram where the adjacent angles are congruent, it means that opposite sides are parallel and congruent, and opposite angles are also congruent. But does that automatically make it a square? Absolutely not, my friend.
A square is a special type of parallelogram where all four sides are equal in length and all four angles are 90 degrees. So, having congruent adjacent angles doesn't necessarily guarantee that we have equal side lengths or right angles.
What about a rhombus then? A rhombus is indeed a parallelogram where all four sides are equal in length, but the angles can vary. Therefore, having congruent adjacent angles doesn't give us enough information to conclude that the quadrilateral is a rhombus.
Now we're left with two options: rectangle and isosceles trapezoid. A rectangle is a special type of parallelogram where all four angles are right angles, which means they are congruent. So, congruent adjacent angles do make the quadrilateral a rectangle.
On the other hand, an isosceles trapezoid is a quadrilateral with only one pair of parallel sides, where the non-parallel sides are congruent, but the angles vary. So, congruent adjacent angles don't necessarily tell us that the quadrilateral is an isosceles trapezoid.
Therefore, my friend, the correct answer is indeed C. Quadrilateral ABCD must be a rectangle if the adjacent angles are congruent. Hope that clears up the confusion and brings a smile to your face!
To determine the correct answer, let's first review the properties of each type of quadrilateral:
A. A square is a special type of parallelogram where all four sides are congruent, and all angles are 90 degrees. However, the given information only states that adjacent angles are congruent, which does not guarantee that all angles are right angles. Therefore, option A is not guaranteed to be true.
B. A rhombus is a parallelogram where all four sides are congruent. Since the given information states that adjacent angles are congruent, it does not guarantee that all sides are congruent. Therefore, option B is not guaranteed to be true.
C. A rectangle is a parallelogram where all angles are 90 degrees. Since the given information states that adjacent angles are congruent, it implies that all angles are congruent and, therefore, all angles are 90 degrees. Therefore, option C is guaranteed to be true.
D. An isosceles trapezoid is a quadrilateral with one pair of opposite sides that are parallel, but the other pair of opposite sides are not parallel. The given information does not provide any details about the parallelism of the sides of the quadrilateral, so option D is not guaranteed to be true.
Based on the given information, option C is the only statement that can be guaranteed to be true: Quadrilateral ABCD is a rectangle.