ABCD is a trapezium in which AB || DC and DC is perpendicular to BC. If ∠DAB = 110°, then ∠ABC - ∠ADC =_____.
20°
Understanding the Trapezium and its Angles
The problem describes a trapezium ABCD with specific properties. A trapezium is a quadrilateral with at least one pair of parallel sides. In this case, we are given that side AB is parallel to side DC (\(AB \parallel DC\)). We are also told that side DC is perpendicular to side BC (\(DC \perp BC\)), which means the angle \(\angle DCB\) is 90 degrees.
We are given the measure of angle \(\angle DAB = 110^\circ\). Our goal is to find the difference between angle \(\angle ABC\) and angle \(\angle ADC\), i.e., \(\angle ABC - \angle ADC\).
Using Properties of Parallel Lines in a Trapezium
When two parallel lines are intersected by a transversal line, the consecutive interior angles (angles on the same side of the transversal and between the parallel lines) are supplementary. In our trapezium ABCD, where \(AB \parallel DC\), we can consider the sides AD and BC as transversals.
Finding Angle ADC
Consider the transversal AD intersecting the parallel lines AB and DC. The angles \(\angle DAB\) and \(\angle ADC\) are consecutive interior angles on the same side of the transversal AD. Therefore, their sum is 180 degrees.
Using the property:
\(\angle DAB + \angle ADC = 180^\circ\)
We are given \(\angle DAB = 110^\circ\).
\(110^\circ + \angle ADC = 180^\circ\)
Subtracting 110° from both sides:
\(\angle ADC = 180^\circ - 110^\circ\)
\(\angle ADC = 70^\circ\)
Finding Angle ABC
Now, consider the transversal BC intersecting the parallel lines AB and DC. The angles \(\angle ABC\) and \(\angle DCB\) are consecutive interior angles on the same side of the transversal BC. Therefore, their sum is 180 degrees.
Using the property:
\(\angle ABC + \angle DCB = 180^\circ\)
We are given that DC is perpendicular to BC, which means \(\angle DCB = 90^\circ\).
\(\angle ABC + 90^\circ = 180^\circ\)
Subtracting 90° from both sides:
\(\angle ABC = 180^\circ - 90^\circ\)
\(\angle ABC = 90^\circ\)
Calculating the Difference
We need to find the value of \(\angle ABC - \angle ADC\).
We found \(\angle ABC = 90^\circ\) and \(\angle ADC = 70^\circ\).
Difference = \(\angle ABC - \angle ADC = 90^\circ - 70^\circ\)
Difference = \(20^\circ\)
Summary of Angles
| Angle | Measure |
|---|---|
| \(\angle DAB\) | \(110^\circ\) (Given) |
| \(\angle DCB\) | \(90^\circ\) (Given, DC \(\perp\) BC) |
| \(\angle ADC\) | \(70^\circ\) (Calculated from \(\angle DAB\)) |
| \(\angle ABC\) | \(90^\circ\) (Calculated from \(\angle DCB\)) |
| \(\angle ABC - \angle ADC\) | \(20^\circ\) (Calculated difference) |
The difference between angle ABC and angle ADC is 20 degrees.
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