Question 17
What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
Hint: Draw a diagonal and check for congruent triangles.
If a quadrilateral has opposite sides equal, we can show it is a parallelogram.
Step 1 — Draw and label
Let us draw a quadrilateral. We will name its vertices A, B, C, and D in order. We are told that its opposite sides are equal in length. This means side AB is equal to side DC. Also, side AD is equal to side BC. Now, let us draw a line segment connecting vertex A to vertex C. This line is called a diagonal. This diagonal splits the quadrilateral into two triangles. These are triangle ABC and triangle CDA.
Step 2 — Check for congruent triangles
Let us compare the two triangles we just formed: and . We know some things about their sides. Side AB is equal to side CD. (This was given in the problem) Side BC is equal to side DA. (This was also given in the problem) Side AC is common to both triangles. So, AC is equal to CA. When three sides of one triangle are equal to the three corresponding sides of another triangle, we say the triangles are congruent. This is called the SSS (Side-Side-Side) congruence rule. So, is congruent to .
Step 3 — Find equal angles
When two triangles are congruent, all their corresponding parts are equal. This is often called CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Let us look at the angles inside our triangles. Since , we know that: Angle BAC is equal to angle DCA. Angle BCA is equal to angle DAC.
Step 4 — Identify parallel lines
Now, let us think about lines and transversals. A transversal is a line that cuts across two or more other lines. Consider lines AB and DC. The diagonal AC acts as a transversal cutting these two lines. We found that angle BAC is equal to angle DCA. These are alternate interior angles. When alternate interior angles formed by a transversal are equal, the two lines are parallel. So, line AB is parallel to line DC.
Next, consider lines AD and BC. The diagonal AC also acts as a transversal cutting these two lines. We found that angle DAC is equal to angle BCA. These are also alternate interior angles. Since these alternate interior angles are equal, the lines AD and BC must be parallel. So, line AD is parallel to line BC.
Step 5 — Name the quadrilateral
We have shown that both pairs of opposite sides of the quadrilateral ABCD are parallel. A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. Therefore, the quadrilateral ABCD is a parallelogram.
Answer
(i) The type of quadrilateral is a Parallelogram.
More questions in FIO
Find all the other angles inside the following rectangles.
Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of
(i) 30° (ii) 40° (iii) 90° (iv) 140°
Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
Find the remaining angles in the following quadrilaterals.
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Find the remaining angles in the following trapeziums—
Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions—
(i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
If PAIR and RODS are two rectangles, find .
Construct a square with diagonal 6 cm without using a protractor.
CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).
If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
Hint: Draw a diagonal and check for congruent triangles.
Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
State whether the following statements are true or false. Justify your answers.
(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
(ii) A quadrilateral having three right angles must be a rectangle.
(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.
(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.
(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.
(vi) A quadrilateral in which all the angles are equal is a rectangle.
(vii) Isosceles trapeziums are parallelograms.