Question

what is the measure of ∠adc in quadrilateral abcd? 45° 65° 115° 135°

Explanation:

Step1: Identify the quadrilateral type

The quadrilateral \(ABCD\) appears to be a parallelogram (since \(AB \parallel CD\) and \(AD \parallel BC\) as per the diagram's structure). In a parallelogram, consecutive angles are supplementary, and opposite angles are equal. Also, consecutive angles (like \(\angle A\) and \(\angle D\)) are supplementary? Wait, no—wait, in a parallelogram, adjacent angles (consecutive angles) are supplementary? Wait, no, actually, in a parallelogram, consecutive angles are supplementary? Wait, no, let's correct: In a parallelogram, opposite sides are parallel, so consecutive angles (e.g., \(\angle A\) and \(\angle B\)) are supplementary, but \(\angle A\) and \(\angle D\): Wait, no, looking at the diagram, \(AD\) and \(BC\) are horizontal, \(AB\) and \(CD\) are slanting. So \(AB \parallel CD\) and \(AD \parallel BC\), so it's a parallelogram. In a parallelogram, consecutive angles (angles adjacent to each other, sharing a side) are supplementary. Wait, \(\angle A\) and \(\angle D\): Do they share a side? \(AD\) is a side, so \(\angle A\) and \(\angle D\) are adjacent angles (consecutive) along side \(AD\). Wait, no—wait, in a parallelogram, adjacent angles (like \(\angle A\) and \(\angle B\)) are supplementary, but \(\angle A\) and \(\angle D\): Wait, maybe I made a mistake. Wait, in a parallelogram, opposite angles are equal, and consecutive angles (adjacent) are supplementary. Wait, let's check the sides: \(AB \parallel CD\) and \(AD \parallel BC\). So \(AD\) is a transversal cutting \(AB\) and \(CD\), so \(\angle A\) and \(\angle D\) are same - side interior angles? Wait, no, \(AB \parallel CD\), and \(AD\) is a transversal, so \(\angle A\) and \(\angle D\) are same - side interior angles, which are supplementary? Wait, no, same - side interior angles are supplementary when lines are parallel. Wait, \(AB \parallel CD\), and \(AD\) is a transversal, so \(\angle A\) and \(\angle D\) are same - side interior angles, so \(\angle A+\angle D = 180^{\circ}\)? But wait, the diagram shows \(\angle A = 115^{\circ}\). Wait, that can't be. Wait, maybe it's a parallelogram where \(AD\) and \(BC\) are parallel, and \(AB\) and \(CD\) are parallel. So \(AD\) is parallel to \(BC\), and \(AB\) is parallel to \(CD\). So angle at \(A\) (\(\angle BAD\)) and angle at \(D\) (\(\angle ADC\)): Wait, maybe I misread the diagram. Wait, the diagram has points \(A\), \(D\) on the bottom (horizontal), \(B\) on the left, \(C\) on the right. So \(AB\) connects \(A\) to \(B\), \(BC\) connects \(B\) to \(C\), \(CD\) connects \(C\) to \(D\), \(AD\) connects \(A\) to \(D\). So \(AD\) is horizontal, \(BC\) is horizontal (so \(AD \parallel BC\)), and \(AB\) and \(CD\) are slanting (so \(AB \parallel CD\)). Therefore, \(ABCD\) is a parallelogram. In a parallelogram, consecutive angles (e.g., \(\angle A\) and \(\angle B\)) are supplementary, but \(\angle A\) and \(\angle D\): Wait, no, \(\angle A\) and \(\angle D\) are adjacent angles along \(AD\)? Wait, no, \(\angle A\) is between \(AB\) and \(AD\), \(\angle D\) is between \(AD\) and \(CD\). Since \(AB \parallel CD\), the consecutive angles \(\angle A\) and \(\angle D\) are same - side interior angles, so they should be supplementary? Wait, but that would mean \(\angle D=180 - 115=65^{\circ}\)? But wait, maybe I got the direction wrong. Wait, no—wait, in a parallelogram, opposite angles are equal, and consecutive angles (adjacent) are supplementary. Wait, \(\angle A\) and \(\angle C\) are opposite, \(\angle B\) and \(\angle D\) are opposite? No, wait, in a parallelogram, opposite angles are equal. So \(\angle A=\angle C\), \(\angle B = \angle D\). And consecutive angles (e.g., \(\angle A\) and \(\angle B\)) are supplementary. Wait, maybe the diagram is a parallelogram where \(AD\) is parallel to \(BC\), so \(\angle A\) and \(\angle B\) are supplementary, \(\angle B\) and \(\angle C\) are supplementary, etc. Wait, maybe I made a mistake. Wait, let's look at the options. The angle at \(A\) is \(115^{\circ}\). If \(ABCD\) is a parallelogram, then \(\angle A\) and \(\angle D\) are adjacent angles (since \(AD\) is a side), and in a parallelogram, adjacent angles are supplementary? Wait, no, adjacent angles (sharing a common side) in a parallelogram are supplementary. Wait, \(AB \parallel CD\), and \(AD\) is a transversal, so \(\angle A\) and \(\angle D\) are same - side interior angles, so they are supplementary. So \(\angle A+\angle D = 180^{\circ}\). So \(\angle D=180 - 115 = 65^{\circ}\)? But wait, the options include \(65^{\circ}\) and \(115^{\circ}\). Wait, maybe I got the sides wrong. Wait, maybe \(AD\) and \(BC\) are parallel, and \(AB\) and \(CD\) are parallel, so \(\angle A\) and \(\angle D\) are not supplementary but \(\angle A=\angle D\)? No, that can't be. Wait, maybe the quadrilateral is a parallelogram, and \(\angle A\) and \(\angle D\) are consecutive angles? Wait, no, let's re - examine the diagram. The points are \(A\), \(D\) on the bottom (so \(AD\) is a horizontal side), \(B\) above \(A\), \(C\) above \(D\). So \(AB\) is from \(A\) to \(B\), \(BC\) is from \(B\) to \(C\) (horizontal), \(CD\) is from \(C\) to \(D\), \(AD\) is from \(A\) to \(D\) (horizontal). So \(AD \parallel BC\) (both horizontal), and \(AB \parallel CD\) (both slanting). So it's a parallelogram. In a parallelogram, opposite angles are equal, and consecutive angles (e.g., \(\angle A\) and \(\angle B\)) are supplementary. Wait, \(\angle A\) is at \(A\), between \(AB\) and \(AD\); \(\angle D\) is at \(D\), between \(AD\) and \(CD\). Since \(AB \parallel CD\), the alternate interior angles would be equal, but no—wait, \(AD\) is a transversal for \(AB\) and \(CD\), so \(\angle BAD\) (angle at \(A\)) and \(\angle CDA\) (angle at \(D\)) are same - side interior angles, so they are supplementary. Therefore, \(\angle ADC=180^{\circ}-\angle BAD = 180 - 115=65^{\circ}\). Wait, but let's check the options. The options are \(45^{\circ}\), \(65^{\circ}\), \(115^{\circ}\), \(135^{\circ}\). So \(65^{\circ}\) is an option. Alternatively, maybe I made a mistake and \(\angle A\) and \(\angle D\) are equal? But that would be if they are opposite angles, but in a parallelogram, opposite angles are equal. Wait, \(\angle A\) and \(\angle C\) are opposite, \(\angle B\) and \(\angle D\) are opposite. So if \(\angle A = 115^{\circ}\), then \(\angle C=115^{\circ}\), and \(\angle B=\angle D\). And since the sum of angles in a quadrilateral is \(360^{\circ}\), \(2\times115 + 2\times\angle D=360\), so \(230+2\angle D = 360\), \(2\angle D=130\), \(\angle D = 65^{\circ}\). Yes, that's correct. So the measure of \(\angle ADC\) is \(65^{\circ}\).

Answer:

\(65^{\circ}\) (corresponding to the option with \(65^{\circ}\))