Question

what is the measure of angle z? 124° 71° degrees

Explanation:

Step1: Find adjacent angles

First, find the adjacent angles to the triangle's angles on the straight line. For the \(124^\circ\) angle, its adjacent angle inside the triangle is \(180^\circ - 124^\circ = 56^\circ\). For the \(71^\circ\) angle, its adjacent angle inside the triangle is \(180^\circ - 71^\circ = 109^\circ\)? Wait, no, wait. Wait, the sum of angles in a triangle is \(180^\circ\), and also, a straight line is \(180^\circ\). Wait, actually, the two angles on the straight line with the triangle's angles: the angle adjacent to \(124^\circ\) is \(180 - 124 = 56^\circ\), and the angle adjacent to \(71^\circ\) is \(180 - 71 = 109^\circ\)? No, that can't be. Wait, no, the triangle has three angles. Let's correct: the straight line is \(180^\circ\), so the two angles adjacent to the triangle's base angles are supplementary. Wait, actually, the triangle's two base angles (the ones on the straight line) and the third angle (angle Z) should satisfy: the sum of the two non-adjacent angles? Wait, no. Wait, the exterior angle theorem? Wait, no, let's use the fact that the sum of angles in a triangle is \(180^\circ\), and the sum of angles on a straight line is \(180^\circ\). So first, find the two angles inside the triangle. The angle adjacent to \(124^\circ\) is \(180 - 124 = 56^\circ\). The angle adjacent to \(71^\circ\) is \(180 - 71 = 109^\circ\)? No, that's not right. Wait, no, the triangle has three angles. Wait, maybe I made a mistake. Wait, the two angles on the straight line: one is \(124^\circ\), the other is \(71^\circ\), but the triangle's two angles are adjacent to these. Wait, actually, the sum of the three angles in the triangle is \(180^\circ\). The two angles inside the triangle (let's call them \(A\) and \(B\)) and angle \(Z\) sum to \(180^\circ\). Also, \(A + 124^\circ = 180^\circ\) (since they are on a straight line), so \(A = 180 - 124 = 56^\circ\). Similarly, \(B + 71^\circ = 180^\circ\)? No, wait, no, the \(71^\circ\) is an exterior angle? Wait, no, the diagram shows a triangle with a base on a straight line, with one exterior angle \(124^\circ\) and another exterior angle \(71^\circ\)? Wait, no, looking at the diagram: the straight line is the base, and the triangle is above it. So the two angles at the base of the triangle (on the straight line) have exterior angles: one is \(124^\circ\), the other is \(71^\circ\). Wait, no, the \(124^\circ\) is an exterior angle, and \(71^\circ\) is another exterior angle? No, the straight line is a straight angle, so the sum of the two interior angles at the base and their exterior angles should be \(180^\circ\) each. Wait, let's start over.

Let’s denote the three angles of the triangle as \(Z\), \(A\), and \(B\), where \(A\) is adjacent to the \(124^\circ\) angle, and \(B\) is adjacent to the \(71^\circ\) angle. Since \(A\) and \(124^\circ\) are supplementary (they form a straight line), \(A = 180^\circ - 124^\circ = 56^\circ\). Similarly, \(B\) and \(71^\circ\) are supplementary? Wait, no, the \(71^\circ\) is an angle on the straight line, so \(B + 71^\circ = 180^\circ\)? No, that would mean \(B = 109^\circ\), but then the sum of \(A + B + Z = 56 + 109 + Z = 165 + Z = 180\), so \(Z = 15\), which doesn't make sense. Wait, I must have messed up the diagram. Wait, looking at the diagram: the straight line has two angles: one is \(124^\circ\) (on the left), then the triangle's left angle, then the triangle's base, then the triangle's right angle, then \(71^\circ\) (on the right). So the left angle of the triangle is adjacent to \(124^\circ\), so they are supplementary: left angle \(= 180 - 124 = 56^\circ\). The right angle of the triangle is adjacent to \(71^\circ\), so they are supplementary: right angle \(= 180 - 71 = 109^\circ\)? No, that can't be, because the sum of angles in a triangle can't have two angles summing to \(56 + 109 = 165\), so \(Z = 15\), but that seems odd. Wait, maybe the \(71^\circ\) is an interior angle? Wait, no, the diagram shows the \(71^\circ\) on the straight line, so it's an exterior angle? Wait, no, maybe I misread the diagram. Wait, perhaps the two angles on the straight line are \(124^\circ\) and \(71^\circ\), but the triangle is between them, so the two angles inside the triangle are \(180 - 124 = 56^\circ\) and \(180 - 71 = 109^\circ\)? No, that's impossible. Wait, no, the sum of angles in a triangle is \(180^\circ\), so maybe the \(71^\circ\) is an interior angle? Wait, the problem's diagram: the triangle has angle \(Z\) at the top, and the base is on a straight line. On the left, there's a \(124^\circ\) angle (exterior), and on the right, a \(71^\circ\) angle (exterior). Wait, no, maybe the \(71^\circ\) is an interior angle. Wait, let's check the exterior angle theorem: the exterior angle is equal to the sum of the two non-adjacent interior angles. So if \(124^\circ\) is an exterior angle, then it should equal \(Z + 71^\circ\). So \(Z = 124 - 71 = 53^\circ\). Ah! That makes sense. So the exterior angle theorem: an exterior angle of a triangle is equal to the sum of the two remote interior angles. So here, the \(124^\circ\) angle is an exterior angle, and the two remote interior angles are \(Z\) and \(71^\circ\). Wait, no, wait: the exterior angle is formed by extending one side, so the exterior angle at the left would be equal to the sum of the two non-adjacent interior angles (angle \(Z\) and the right interior angle). Wait, maybe the right angle is \(71^\circ\)? No, the diagram shows \(71^\circ\) on the straight line. Wait, let's re-express:

Let’s consider the straight line: the sum of angles on a straight line is \(180^\circ\). The triangle has three angles: \(Z\), \(A\) (left base angle), \(B\) (right base angle). Then, \(A + 124^\circ = 180^\circ\) (since they are supplementary), so \(A = 56^\circ\). \(B + 71^\circ = 180^\circ\)? No, that would be \(B = 109^\circ\), but then \(A + B + Z = 56 + 109 + Z = 165 + Z = 180\), so \(Z = 15\), which is wrong. So clearly, my initial assumption is wrong.

Wait, maybe the \(71^\circ\) is an interior angle. Let's try that. If the right base angle is \(71^\circ\), and the left exterior angle is \(124^\circ\), then the left base angle is \(180 - 124 = 56^\circ\). Then, the sum of angles in the triangle: \(56 + 71 + Z = 180\). So \(56 + 71 = 127\), so \(Z = 180 - 127 = 53^\circ\). Ah! That makes sense. So the \(71^\circ\) is an interior angle, not an exterior angle. So the diagram has the right base angle as \(71^\circ\), and the left exterior angle as \(124^\circ\), so the left base angle is \(180 - 124 = 56^\circ\). Then, sum of angles: \(56 + 71 + Z = 180\). So \(Z = 180 - 56 - 71 = 53^\circ\).

Step2: Calculate angle Z

Now, sum the two interior base angles: \(56^\circ + 71^\circ = 127^\circ\). Then, angle \(Z = 180^\circ - 127^\circ = 53^\circ\).

Answer:

\(53\)