Answer the following with the response of True or False:
1) A trapezoid has one pair of parallel sides. ____________________
True
2) In an isosceles trapezoid, the non-parallel sides are congruent. ____________________
True
3) In a rhombus, all sides are congruent. ____________________
True
4) In a rectangle, diagonals are perpendicular. ____________________
False
In square and in rhombus, the diagonals are perpendicular. For Parallelogram, rectangle, and trapeziums, the diagonals are not perpendicular.
5) In an isosceles trapezoid, opposite angles are congruent. ____________________
False
The opposite angles are supplementary.
6) All quadrilaterals are rectangles. ____________________
False
Rectangle can be considered as quadrilateral but all quadrilateral not not rectangles.
Problem 1 :
Which of the following is true about every parallelogram ?
a) All four sides are congruent
b) The diagonals are perpendicular to each other.
c) Two pairs of opposite angles are congruent
d) The consecutive angles are congruent.
Solution :
In parallelogram, opposite sides are parallel and equal.
In triangle ADC and in ABC,
∠DCA = ∠BAC (DC || AB, alternate interior angles)
∠DAC = ∠BCA (AD || BC, alternate interior angles)
AC = AC (Common)
So, triangles ADC and ABC are congruent. Using CPCT
∠D = ∠B
Opposite angles are equal.
Problem 2 :
Which reason can be used to prove that a parallelogram is a rhombus ?
a) The diagonals are congruent
b) The opposite angles are congruent
c) The diagonals are perpendicular
d) The opposite sides are parallel.
Solution :
In rhombus, the diagonal will be perpendicular. But in parallelogram the diagonals will not be perpendicular.
Problem 3 :
For which quadrilateral are the diagonals congruent but do not bisect each other ?
a) Parallelogram
b) Isosceles trapezoid
c) Rectangle
d) Rhombus
Solution :
In isosceles trapezoid, the diagonals are congruent but they will not bisect each other.
Problem 4 :
For which quadrilaterals are the diagonals congruent ?
a) Rhombus b) Parallelogram c) Isosceles trapezoid
d) Rectangle e) Square
Solution :
In Isosceles trapezoid, Rectangle and Square the diagonals are congruent.
Problem 5 :
If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral could be a
a) Trapezoid b) Square c) Rectangle d) Rhombus
Solution :
In trapezoid, the diagonals do not bisect each other.
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