Question

four triangles are shown. one side of each triangle lies on a ray, and the triangles are not drawn to scale. based on these triangles, which statement is true? a = 298, because 53 + 65 = 118 and 118 + 180 = 298 a = 168, because 180 - 65 = 115 and 115 + 53 = 168 a = 62, because 53 + 65 = 118 and 180 - 118 = 62 a = 118, because 180 - (53 + 65) = 62 and 180 - 62 = 118

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. Let's assume the three interior angles of the triangle are \(x\), \(y\), and \(z\), so \(x + y+z = 180^{\circ}\). If we know two angles, say \(x = 53^{\circ}\) and \(y = 65^{\circ}\), we can find the third - interior angle.

Step2: Calculate the third - interior angle

Let the third - interior angle be \(z\). Then \(z=180-(53 + 65)=180 - 118 = 62^{\circ}\).

Step3: Use the linear - pair property

The angle \(a\) forms a linear pair with the third - interior angle of the triangle. A linear pair of angles sums to 180°. So \(a = 180 - 62=118^{\circ}\).

Answer:

The correct statement is \(a = 118\), because \(180-(53 + 65)=62\) and \(180 - 62=118\)