Quadrilaterals

A quadrilateral is a two-dimensional closed shape that has four straight sides.

Three special quadrilaterals are a parallelogram which is a quadrilateral with opposite sides parallel and congruent, a trapezoid which is a quadrilateral that has one pair of opposite sides parallel and a kite which is a quadrilateral that has two pairs of congruent adjacent sides.

Three special parallelograms are a rectangle which has every angle a right angle and opposite sides both congruent and parallel, a square which has every angle a right angle, opposite sides parallel and all sides congruent and a rhombus which has every side equal in length, opposite sides parallel, but not every angle is a right angle.

Remember a quadrilateral is a two-dimensional closed shape that has four straight sides and four angles.

The sum of the interior angles of a quadrilateral is 360° regardless of the type of quadrilateral. The following diagram will show various quadrilaterals and their interior angles.

If you add the measure of the four interior angles of each quadrilateral shown above, the sum will equal 360°. Each of these quadrilaterals can be divided into two triangles. Remember the sum of the interior angles of a triangle equals 180°. If the closed figure contains two triangles then the sum of the interior angles equals 2(180∘)=360∘.

Let’s apply these facts to an example.

For the following quadrilateral, determine the measure of  ∠M.

First, write an equation to represent the sum of the interior angles of quadrilateral JKLM.

Next, substitute into the equation, the given measures of angles J, K and L. 

Next, simplify the left side of the equation.

The measure of ∠M is 65°.

This method can be used to calculate the measure of the missing angle of any quadrilateral when the measures of three of the angles are known.

Prove that the sum of the angles of a quadrilateral is 360°.

To prove the required property we use the property of the sum of interior angles of a triangle.

• Sum of interior angles of a triangle is equal to 180∘.

Step 1:

Draw a quadrilateral ABCD

Sides of quadrilateral ABCD are: AB, BC, CD, DA

Angles of quadrilateral ABCD are: ∠A, ∠B, ∠C, ∠D

Step 2:

Join vertices A and C

Join the opposite vertices A and C to divide the quadrilateral ABCD into two halves.

Quadrilateral ABCD is divided into two triangles, △ABC and △ADC.

⇒ Sum of angles of quadrilateral

    ABCD = ∠CAD + ∠CAB + ∠B + ∠BCA + ∠DCA + ∠D    ——— (2)

Thus, sum of angles of a quadrilateral is 360∘

Example

1. Three angles of a quadrilateral are 54°, 80° and 116°. Find the measure of the fourth angle. 

Solution:

Let the measure of the fourth angle be x°.

We know that the sum of the angles of a quadrilateral is 360°. 

Therefore, 54 + 80 + 116 + x = 360 

⇒ 250 + x = 360

⇒ x = (360 – 250) = 110.

Hence, the measure of the fourth angle is 110°.

2. The three angles of the quadrilateral are 90°, 105°, 85°. Find the measure of the fourth angle of a quadrilateral.

Solution:

We know that sum of all the angles of a quadrilateral is 360°.

Let the unknown angle of the quadrilateral be x.

Then 90° + 105° + 85° + x = 360°

⇒ 280° + x = 360°  

⇒ x = 360 – 280

⇒ x = 80°

Therefore, the measure of the fourth angle of the quadrilateral is 80°

3. The measures of two angles of a quadrilateral are 115°and 45°, and the other two angles are equal. Find the measure of each of the equal angles.

Solution:

Let the measure of each of the equal angles be x°.

We know that the sum of all the angles of a quadrilateral is 360°.

Therefore, 115 + 45 + x + x = 360

⇒ 160 + 2x = 360

⇒ 2x = (360 – 160) = 200

⇒ x = 100.

Hence, the measure of each of the equal angles is 100°.