Question

∠1
∠2
a b
80° 70°
d 1 2 c

Explanation:

Step1: Identify the figure type

The figure \(ABCD\) has \(AB \parallel DC\) and \(AD\) and \(BC\) as non - parallel sides (from the direction of arrows, \(AB\) and \(DC\) are parallel). So, \(ABCD\) is a trapezoid with \(AB\parallel DC\).

Step2: Use the property of consecutive angles in a trapezoid

In a trapezoid, consecutive angles between the two parallel sides are supplementary (sum to \(180^{\circ}\)).

For \(\angle1\) and \(\angle A\): Since \(AB\parallel DC\) and \(AD\) is a transversal, \(\angle1+\angle A = 180^{\circ}\). Given \(\angle A=80^{\circ}\), then \(\angle1 = 180^{\circ}-\angle A=180 - 80=100^{\circ}\).

For \(\angle2\) and the angle at \(B\) (which is \(70^{\circ}\) with the exterior angle, so the interior angle at \(B\) is \(180 - 70 = 110^{\circ}\)? Wait, no. Wait, the angle at \(B\) with the exterior angle of \(70^{\circ}\): the interior angle at \(B\) and the \(70^{\circ}\) angle are supplementary. So interior angle at \(B=180 - 70=110^{\circ}\)? Wait, no, wait the arrows: \(AB\) and \(DC\) are parallel, so the angle at \(B\) (interior) and \(\angle2\) are supplementary. Wait, no, let's re - examine.

Wait, the angle at \(B\) has an exterior angle of \(70^{\circ}\), so the interior angle at \(B\) is \(180 - 70=110^{\circ}\)? No, wait the diagram: \(AB\) is a horizontal line (with arrow to the right), \(BC\) makes an angle with \(AB\) such that the exterior angle is \(70^{\circ}\). So the interior angle at \(B\) (between \(AB\) and \(BC\)) is \(180 - 70 = 110^{\circ}\). But since \(AB\parallel DC\), \(\angle2+\) interior angle at \(B=180^{\circ}\)? No, wait \(AB\parallel DC\), so \(\angle A+\angle1 = 180^{\circ}\) (same - side interior angles) and \(\angle B+\angle2=180^{\circ}\) (same - side interior angles).

Wait, \(\angle A = 80^{\circ}\), so \(\angle1=180 - 80 = 100^{\circ}\). The angle at \(B\): the exterior angle is \(70^{\circ}\), so the interior angle at \(B\) is \(180 - 70=110^{\circ}\)? No, wait the exterior angle at \(B\) is \(70^{\circ}\), so the interior angle at \(B\) is \(180 - 70 = 110^{\circ}\). Then since \(AB\parallel DC\), \(\angle2 +\) interior angle at \(B=180^{\circ}\)? No, that can't be. Wait, maybe I made a mistake. Wait the angle at \(B\): the angle between \(AB\) (going to the right) and \(BC\) (going down to \(C\)): the exterior angle is \(70^{\circ}\) (the angle outside the trapezoid, formed by extending \(AB\) or \(BC\)). Wait, actually, the angle at \(B\) (interior) and the \(70^{\circ}\) angle are adjacent and supplementary. So interior angle at \(B = 180 - 70=110^{\circ}\). But since \(AB\parallel DC\), \(\angle A\) and \(\angle1\) are same - side interior angles, so \(\angle A+\angle1 = 180^{\circ}\), so \(\angle1=100^{\circ}\). \(\angle B\) and \(\angle2\) are same - side interior angles, so \(\angle B+\angle2 = 180^{\circ}\), so \(\angle2=180 - 110 = 70^{\circ}\)? Wait, no, that contradicts. Wait, maybe the angle at \(B\) is \(70^{\circ}\) as an interior angle? Wait, the diagram shows at \(B\), the angle with the arrow - the exterior angle is \(70^{\circ}\), but maybe I misread. Let's start over.

Correct approach: \(AB\parallel DC\) (given by the arrows). So, \(\angle A\) and \(\angle1\) are same - side interior angles, so \(\angle A+\angle1 = 180^{\circ}\). Given \(\angle A = 80^{\circ}\), so \(\angle1=180 - 80 = 100^{\circ}\).

For \(\angle2\): The angle at \(B\) has an exterior angle of \(70^{\circ}\), so the interior angle at \(B\) (between \(AB\) and \(BC\)) is \(180 - 70=110^{\circ}\)? No, wait the angle at \(B\) (the angle between \(AB\) and \(BC\)) and \(\angle2\) are same - side interior angles (since \(AB\parallel DC\) and \(BC\) is a transversal). So \(\angle B+\angle2 = 180^{\circ}\). But what is \(\angle B\)? The angle at \(B\) and the \(70^{\circ}\) angle: the \(70^{\circ}\) angle is adjacent to \(\angle B\) and they are supplementary (linear pair). So \(\angle B=180 - 70 = 110^{\circ}\). Then \(\angle2=180 - 110 = 70^{\circ}\)? Wait, no, that can't be. Wait, maybe the \(70^{\circ}\) is the interior angle? Wait, the diagram: \(AB\) is horizontal (arrow right), \(BC\) goes down to \(C\), and the angle between \(BC\) and the extension of \(AB\) is \(70^{\circ}\). So the interior angle at \(B\) is \(180 - 70 = 110^{\circ}\). Then since \(AB\parallel DC\), \(\angle2+\angle B = 180^{\circ}\), so \(\angle2 = 180 - 110=70^{\circ}\).

Wait, let's check again.

\(\angle A = 80^{\circ}\), \(AB\parallel DC\), \(AD\) is transversal: \(\angle1+\angle A=180^{\circ}\Rightarrow\angle1 = 100^{\circ}\).

\(\angle B\) has an exterior angle of \(70^{\circ}\), so interior angle at \(B = 180 - 70=110^{\circ}\). \(AB\parallel DC\), \(BC\) is transversal: \(\angle2+\angle B = 180^{\circ}\Rightarrow\angle2=180 - 110 = 70^{\circ}\).

Answer:

\(\angle1 = 100^{\circ}\), \(\angle2=70^{\circ}\)