Question

7
find the missing angle
triangle with 40° and 17° interior angles, angle ( a^circ ) as an exterior angle at the top vertex
options: 40 (orange), 17 (blue), 123 (green), 57 (red)

Explanation:

Step1: Find the third angle inside the triangle

The sum of the interior angles of a triangle is \(180^\circ\). So, the third angle (let's call it \(x\)) inside the triangle is \(180 - 40 - 17\).
\[180 - 40 - 17 = 123^\circ\]

Step2: Find the angle \(a\)

Angle \(a\) and the \(123^\circ\) angle are supplementary (they form a linear pair), so their sum is \(180^\circ\). Wait, no—actually, angle \(a\) is an exterior angle? Wait, no, looking at the diagram, angle \(a\) and the third interior angle are supplementary? Wait, no, maybe I made a mistake. Wait, the two given angles are \(40^\circ\) and \(17^\circ\), so the third interior angle is \(180 - 40 - 17 = 123^\circ\)? Wait, no, that can't be, because then angle \(a\) would be supplementary to that? Wait, no, maybe the diagram shows that angle \(a\) is an exterior angle, but actually, the sum of the interior angles of a triangle is \(180^\circ\), so the third interior angle is \(180 - 40 - 17 = 123^\circ\)? Wait, no, \(40 + 17 = 57\), so \(180 - 57 = 123\). Then angle \(a\) and that \(123^\circ\) angle are supplementary? Wait, no, maybe the diagram is such that angle \(a\) is adjacent to the third interior angle, forming a linear pair. So angle \(a + 123^\circ = 180^\circ\)? Wait, no, that would make \(a = 57^\circ\), but that contradicts. Wait, no, maybe I got the diagram wrong. Wait, the problem is to find angle \(a\), which is an exterior angle? Wait, no, the two angles inside the triangle are \(40^\circ\) and \(17^\circ\), so the third interior angle is \(180 - 40 - 17 = 123^\circ\). Then angle \(a\) is supplementary to that? Wait, no, maybe the diagram is a triangle with two angles \(40^\circ\) and \(17^\circ\), and angle \(a\) is the exterior angle adjacent to the third interior angle. Wait, the exterior angle theorem says that the exterior angle is equal to the sum of the two non - adjacent interior angles. So angle \(a = 40 + 17 = 57^\circ\)? Wait, that makes sense. Let's re - calculate.

The sum of the interior angles of a triangle is \(180^\circ\). Let the third interior angle be \(y\). Then \(y=180-(40 + 17)=180 - 57 = 123^\circ\). Now, angle \(a\) and angle \(y\) are supplementary (they form a straight line), so \(a + y=180^\circ\), so \(a = 180 - 123 = 57^\circ\)? Wait, no, that's not right. Wait, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So if angle \(a\) is an exterior angle, then \(a = 40+17 = 57^\circ\). Wait, maybe the diagram shows that angle \(a\) is an exterior angle, so we can use the exterior angle theorem. So the two non - adjacent interior angles are \(40^\circ\) and \(17^\circ\), so \(a = 40 + 17=57^\circ\). Wait, but earlier when we calculated the third interior angle as \(123^\circ\), and then \(a = 180 - 123 = 57^\circ\), which matches. So the correct value of \(a\) is \(57^\circ\)? Wait, no, wait \(40+17 = 57\), and \(180 - 57 = 123\), then \(180 - 123 = 57\). So angle \(a\) is \(57^\circ\)? Wait, but the options are 40, 17, 123, 57. So the answer should be 57? Wait, no, wait the third interior angle is \(123^\circ\), and angle \(a\) is supplementary to it? Wait, no, maybe the diagram is drawn such that angle \(a\) is the exterior angle, so the answer is \(40 + 17=57\).

Wait, let's start over. The sum of the interior angles of a triangle is \(180^\circ\). So the three interior angles add up to \(180^\circ\). We know two angles: \(40^\circ\) and \(17^\circ\). Let the third interior angle be \(x\). Then \(x=180-(40 + 17)=180 - 57 = 123^\circ\). Now, angle \(a\) and angle \(x\) are adjacent and form a linear pair (they are on a straight line), so their sum is \(180^\circ\). So \(a + x=180^\circ\), so \(a = 180 - x=180 - 123 = 57^\circ\). So the measure of angle \(a\) is \(57^\circ\).

Answer:

\(57^\circ\) (corresponding to the red option with 57)