How to Find the Third Angle in Any Quadrilateral
Introduction to Angles of a Quadrilateral
Sum of the angles of the quadrilateral is \[\left( {n - 2} \right) \times 180^\circ \], the formula for the interior angle sum of a quadrilateral, where n is the number of sides of the provided polygon. 180° - interior angle is the exterior angle. When a quadrilateral's inner angle is known and the value of the matching exterior angle is needed, this formula is utilized.
Definition of a Quadrilateral
A closed quadrilateral has four sides, four vertices, and four angles. It is a form of a polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.
Three Angles of a Quadrilateral Sum
Four-sided polygons with four vertices and four internal angles are known as quadrilaterals. Quadrilaterals come in a variety of shapes, including squares, rectangles, rhombus, and others. Because the internal angles of a quadrilateral always add up to \[360^\circ \], the unknown angles of a quadrilateral can be simply determined if the other angles are known.
Quadrilateral Angles Formula
The number of triangles that can be produced in a polygon can be used to determine the interior angle sum, in accordance with the polygon's angle sum attribute. By tracing diagonal lines from a single vertex, these triangles are created. This can be determined using a formula that states that if a polygon has 'n' sides, it will contain \[\left( {n - 2} \right)\] triangles.
\[S = [n - 2] \times 180\] is a formula that can be used to compute the sum of interior angles in a polygon, where n is the number of sides in the supplied polygon.
When we apply the formula to a quadrilateral, for instance, we get the following results: \[S = [n - 2] \times 180^\circ ,S = [4 - 2] \times 180^\circ = 360^\circ \]. The sum of a quadrilateral's internal angles is therefore always 360 degrees, according to the quadrilateral's angle sum condition. This characteristic aids in determining a quadrilateral's unknown angles. The value of the unknown angle can be determined by subtracting the sum of the other angles from \[360^\circ \] if the other angles are known.
Interior Angle of a Quadrilateral
Its internal angles are the angles that make up a quadrilateral. A quadrilateral's internal angles add up to 360°. This aids in figuring out a quadrilateral's unknown angles. We know that all of the quadrilateral's inner angles are 90 degrees each if it is a square or rectangle.
Exterior Angle of a Quadrilateral
The external angles of a quadrilateral are those that are created between a side and a line that extends from a neighboring side. If we look at the above illustration, we can see that the outer and interior angles combine to form a straight line, which is why they form a linear pair.
Quadrilateral Angle Calculator
Conclusion
The sum of the internal angles of a quadrilateral is 360 degrees. Quadrilaterals are four-sided polygons that have four internal angles and four vertices. Numerous shapes, such as squares, rectangles, and rhombi, can be used to create quadrilaterals.
Solved Examples
Example 1: Find the fourth internal angle of a quadrilateral if the first three are provided as \[77^\circ ,98^\circ and110^\circ \].
Ans: The following formula can be used to get the quadrilateral's fourth angle:
360 - (Sum of the other 3 inner angles) (Sum of the other 3 interior angles)
Fourth angle unknown = 360 - (Sum of the other 3 interior angles)
Unknown fourth angle equals \[360^\circ - \left( {77 + 98 + 110} \right)\]
Fourth angle: \[360^\circ - \left( {77 + 98 + 110} \right] = 75^\circ \]
Example 2: If the inner angle of a quadrilateral is 68°, what is the external angle?
Ans: We are aware that a quadrilateral's interior and exterior angles form a linear pair. We will find the answer via the quadrilateral exterior angle formula.
\[180 - 68 = 112^\circ \] is the outside angle. The external angle is \[112^\circ \] as a result.
Example 3: Find the Missing Angle of a Quadrilateral:
Ans: Applying Angle Sum Property in Quadrilateral we will get:
\[\begin{array}{l}x + 77 + 67 + 101 = 360\\x + 245 = 360\\x = 115\end{array}\]
FAQs on Three Angles of a Quadrilateral Explained
1. What is a quadrilateral and what are its interior angles?
A quadrilateral is a polygon with four sides, four vertices, and four angles. The interior angles are the angles formed inside the quadrilateral at each of its four vertices. According to the angle sum property of quadrilaterals, the sum of these four interior angles is always 360 degrees.
2. If three angles of a quadrilateral are known, how can the fourth angle be found?
To find the fourth angle of a quadrilateral when three angles are given, you use the angle sum property. The sum of all four interior angles in any quadrilateral is 360°. Therefore, you can calculate the fourth angle using the formula:
- Fourth Angle = 360° - (Sum of the three known angles)
3. How can we prove that the sum of the angles in a quadrilateral is 360°?
The proof for the angle sum property of a quadrilateral is a key concept as per the CBSE syllabus. It can be proven by following these steps:
- First, draw any quadrilateral, for example, ABCD.
- Next, draw a diagonal connecting two opposite vertices, such as AC. This divides the quadrilateral into two triangles: △ABC and △ADC.
- We know the sum of angles in any triangle is 180°.
- In △ABC, ∠CAB + ∠B + ∠BCA = 180°.
- In △ADC, ∠DAC + ∠D + ∠DCA = 180°.
- By adding the angles of both triangles, we get the sum of the angles of the quadrilateral: (∠CAB + ∠DAC) + ∠B + (∠BCA + ∠DCA) + ∠D = 180° + 180°.
- This simplifies to ∠A + ∠B + ∠C + ∠D = 360°.
4. How do you find the angles of a quadrilateral if they are in a given ratio?
If the angles of a quadrilateral are given in a ratio, for example, a:b:c:d, you can find each angle with these steps:
- Represent the angles as ax, bx, cx, and dx, where 'x' is a common multiplier.
- Use the angle sum property: ax + bx + cx + dx = 360°.
- Factor out 'x': x(a + b + c + d) = 360°.
- Solve for 'x' by dividing 360° by the sum of the ratio parts: x = 360° / (a + b + c + d).
- Finally, calculate each angle by multiplying the value of 'x' back into ax, bx, cx, and dx.
5. Is it possible for a quadrilateral to have three obtuse angles?
No, a quadrilateral cannot have three obtuse angles. An obtuse angle is an angle greater than 90°. If a quadrilateral had three obtuse angles, the sum of just those three angles would be greater than 270° (for instance, 91° + 91° + 91° = 273°). Since the fourth angle must also have a positive value, the total sum would exceed 360°, which contradicts the fundamental angle sum property of quadrilaterals.
6. How does the angle sum property differ between convex and concave quadrilaterals?
The angle sum property remains the same for both convex and concave quadrilaterals; the sum of the interior angles is always 360°. The key difference lies in the nature of the angles:
- In a convex quadrilateral, all interior angles are less than 180°.
- In a concave quadrilateral, one of the interior angles is a reflex angle (greater than 180°).
7. Why is the sum of angles in a quadrilateral double that of a triangle?
The sum of angles in a quadrilateral (360°) is double that of a triangle (180°) because any quadrilateral can be divided into two non-overlapping triangles by drawing a single diagonal between opposite vertices. Since each triangle contributes 180° to the total sum of interior angles, the combined sum for the quadrilateral is simply 180° + 180° = 360°. This shows a direct geometric relationship between the two shapes.
8. How does knowing the type of a quadrilateral (like a parallelogram or kite) help in finding its angles?
Knowing the type of quadrilateral provides additional rules that simplify finding its angles, which is a key application in geometry. For example:
- In a parallelogram, opposite angles are equal, and consecutive angles add up to 180°. If one angle is known, all others can be found.
- In a kite, one pair of opposite angles is equal.
- In an isosceles trapezium, angles on the same base are equal.
