Question

c. solve for x in each diagram. 1. 2. 3. 4.

Explanation:

Step1: Use exterior - angle property for first triangle

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. For the first triangle with exterior angle $130^{\circ}$, the non - adjacent interior angles are part of a triangle with another angle of $99^{\circ}$. First, find the non - adjacent interior angle to the $130^{\circ}$ exterior angle. Let's call it $y$. We know that $y = 180 - 130=50^{\circ}$. Then, in the triangle with angles $y = 50^{\circ}$, $99^{\circ}$ and $x$, using the angle - sum property of a triangle ($180^{\circ}$ for the sum of interior angles of a triangle), we have $x=180-(50 + 99)=31^{\circ}$.

Step2: Use exterior - angle property for second triangle

For the second triangle, with exterior angle $140^{\circ}$ and one non - adjacent interior angle $132^{\circ}$, using the exterior - angle property of a triangle (exterior angle = sum of non - adjacent interior angles), we have $140=x + 132$. Solving for $x$, we get $x=140 - 132 = 8^{\circ}$.

Step3: Use exterior - angle property for third triangle

For the third triangle, the exterior angle is $81^{\circ}$, and the non - adjacent interior angles are $x$ and $2x$. By the exterior - angle property of a triangle, $81=x + 2x$. Combining like terms, we have $3x=81$. Dividing both sides by 3, we get $x = 27^{\circ}$.

Step4: Use exterior - angle property for fourth triangle

For the fourth triangle, with exterior angle $(x + 8)^{\circ}$, and non - adjacent interior angles $90^{\circ}$ and $64^{\circ}$, using the exterior - angle property of a triangle, we have $x+8=90 + 64$. Then $x=90 + 64-8=146^{\circ}$.

Answer:

1. $x = 31^{\circ}$
2. $x = 8^{\circ}$
3. $x = 27^{\circ}$
4. $x = 146^{\circ}$