How to Understand Angles in Quadrilaterals

Step-by-step Guide: Angles in Quadrilaterals

The Basics of Quadrilaterals:
A quadrilateral, by definition, is a polygon with four sides and four vertices. The sum of its interior angles is always \(360^\circ\).

Types of Quadrilaterals and Their Angles:

• Square: All angles are right angles, each measuring \(90^\circ\).
• Rectangle: All angles are right angles, each \(90^\circ\).
• Rhombus: Opposite angles are equal.
• Parallelogram: Opposite angles are equal; consecutive angles are supplementary (add to \(180^\circ\)).
• Trapezoid: Angles adjacent to the parallel sides are supplementary.

Calculating Missing Angles:
Given three angles in a quadrilateral, the fourth can be found by subtracting the sum of the given three angles from \(360^\circ\).

External Angles:
Every angle in a polygon has an external angle. For quadrilaterals, the sum of the external angles is always \(360^\circ\).

Examples

Example 1:
Finding a Missing Angle in a Parallelogram
In the parallelogram \( ABCD \), we have the following angles:
Angle \( A = 70^\circ \)
Angle \( B = 110^\circ \)
Angle \( C = 70^\circ \)
Angle \( D =? \)

Solution:
To find angle \( D \), we use the property that the sum of interior angles in any quadrilateral is \( 360^\circ \).
Angle \( D = 360^\circ – (70^\circ + 110^\circ + 70^\circ) = 360^\circ – 250^\circ = 110^\circ \).
Thus, angle \( D \) in parallelogram \( ABCD \) is \( 110^\circ \).

Example 2:
Angles in a Trapezoid: In the trapezoid \( WXYZ\), where:
Side \( WX\) is parallel to side \( ZY\).
Angle \( W = 80^\circ \)
Angle \( X = 120^\circ \)
Angle \( Z =? \)
Angle \( Y =? \)

Solution:
For trapezoids, the angles on the same side of the non-parallel line are supplementary. This means they add up to \( 180^\circ \).
Using the supplementary angles property:
Angle \( Y = 180^\circ – 80^\circ = 100^\circ \)
Angle \( Z = 180^\circ – 120^\circ = 60^\circ \).
So, in trapezoid \( WXYZ \), angle \( Y \) is \( 100^\circ \) and angle \( Z \) is \( 60^\circ \).

Practice Questions:

1. In a rectangle, if one angle measures \(85^\circ\), what are the measures of the other three angles?
2. In a rhombus, if one angle is \(130^\circ\), what is the measure of its opposite angle?
3. Given a general quadrilateral with angles \(A\), \(B\), \(C\), and \(D\), if angle \(A = 90^\circ\), angle \(B = 80^\circ\), and angle \(C = 100^\circ\), find angle \(D\).

Answers:

1. In a rectangle, all angles are right angles, each measuring \(90^\circ\). Thus, the given angle is not possible for a rectangle.
2. In a rhombus, opposite angles are equal. Thus, if one angle is \(130^\circ\), its opposite angle is also \(130^\circ\).
3. Angle \(D = 360^\circ – (90^\circ + 80^\circ + 100^\circ) = 90^\circ\).