Question

find the value of x. x = ? 121° 94° 7x°

Explanation:

Step1: Use the exterior angle theorem for polygons (or the sum of exterior angles around a point).

The sum of angles around a point is \( 360^\circ \), but here we can also use the fact that the sum of the exterior angles of a triangle (or the angles formed by the extended sides) can be related. First, find the supplementary angles. The angle adjacent to \( 121^\circ \) is \( 180 - 121 = 59^\circ \), and the angle adjacent to \( 94^\circ \) is \( 180 - 94 = 86^\circ \). But maybe a better way: the sum of the angles around the transversal or using the fact that the sum of the angles in the linear combinations. Wait, actually, the sum of the exterior angles of a triangle (or the angles formed by the extended sides) should relate to the given angles. Wait, the angles given are \( 121^\circ \), \( 94^\circ \), and the angle related to \( 7x \). Wait, actually, the sum of the angles around a point is \( 360^\circ \), but the straight lines are \( 180^\circ \) each. Wait, let's re - examine. The three angles: \( 121^\circ \), \( 94^\circ \), and the angle that is supplementary to \( 7x \) (since \( 7x \) and its adjacent angle form a straight line, \( 180^\circ \)). Wait, no. Let's think of the sum of the angles in the figure. The sum of the angles around the point where the three lines meet (the vertex with the triangle) should satisfy that the sum of the exterior angles or the sum of the angles in the linear pairs. Wait, another approach: the sum of the angles \( 121^\circ+94^\circ + (180 - 7x)^\circ=360^\circ\)? No, that's not right. Wait, actually, the sum of the angles on a straight line is \( 180^\circ \). The two angles \( 121^\circ \) and \( 94^\circ \) and the angle related to \( 7x \) should satisfy: \( 121 + 94+(180 - 7x)=360 \)? Wait, no. Let's calculate the sum of \( 121+94 = 215 \). Then \( 360 - 215=145 \), and \( 180 - 7x = 145 \), so \( 7x=180 - 145 = 35 \), then \( x = 5 \). Wait, let's check again. The sum of the angles around a point is \( 360^\circ \). We have three angles: one is \( 121^\circ \), one is \( 94^\circ \), and the third is \( 180 - 7x \) (because \( 7x \) and this angle form a straight line, so they are supplementary). So \( 121+94+(180 - 7x)=360 \). Calculate \( 121 + 94=215 \), then \( 215+180 - 7x = 360 \), \( 395-7x = 360 \), \( - 7x=360 - 395=-35 \), \( 7x = 35 \), \( x = 5 \).

Step2: Solve for x.

From the equation \( 7x=35 \) (derived from the sum of angles around a point or the sum of linear pairs), we divide both sides by 7: \( x=\frac{35}{7}=5 \).

Answer:

\( x = 5 \)