Practice worksheet for solving for variables in triangle exterior angles using the Exterior Angle Theorem.

To solve for the variables in these problems, we use the Exterior Angle Theorem. This theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles (the two angles inside the triangle that are not touching the exterior angle).

Here is the step-by-step solution for each problem:

1. Solve for $f$
* Exterior angle: $152^\circ$
* Remote interior angles: $34^\circ$ and $7f^\circ$
* Equation: $34 + 7f = 152$
* Subtract 34 from both sides: $7f = 118$
* Divide by 7: $f = \frac{118}{7} \approx 16.86$

2. Solve for $x$
* Exterior angle: $150^\circ$
* Remote interior angles: $3x^\circ$ and $(x + 46)^\circ$
* Equation: $3x + (x + 46) = 150$
* Combine like terms: $4x + 46 = 150$
* Subtract 46 from both sides: $4x = 104$
* Divide by 4: $x = 26$

3. Solve for $d$
* Exterior angle: $125^\circ$
* Remote interior angles: $60^\circ$ and $(d + 20)^\circ$
* Equation: $60 + d + 20 = 125$
* Combine like terms: $d + 80 = 125$
* Subtract 80 from both sides: $d = 45$

4. Solve for $k$
* Exterior angle: $k^\circ$
* Remote interior angles: $28^\circ$ and $(4k - 9)^\circ$
* Equation: $28 + 4k - 9 = k$
* Combine like terms on the left: $4k + 19 = k$
* Subtract $4k$ from both sides: $19 = -3k$
* Divide by -3: $k = -\frac{19}{3} \approx -6.33$
*(Note: While this results in a negative number which is physically impossible for a geometric angle, this is the correct algebraic solution based on the numbers provided in the worksheet.)*

5. Solve for $m$
* Exterior angle: $132^\circ$
* Remote interior angles: $24m^\circ$ and $(8m + 4)^\circ$
* Equation: $24m + 8m + 4 = 132$
* Combine like terms: $32m + 4 = 132$
* Subtract 4 from both sides: $32m = 128$
* Divide by 32: $m = 4$

6. Solve for $g$
* Exterior angle: $90^\circ$
* Remote interior angles: $(13g + 14)^\circ$ and $(6g - 6)^\circ$
* Equation: $13g + 14 + 6g - 6 = 90$
* Combine like terms: $19g + 8 = 90$
* Subtract 8 from both sides: $19g = 82$
* Divide by 19: $g = \frac{82}{19} \approx 4.32$

7. Solve for $b$
* Exterior angle: $(5a + 10)^\circ$ — *Wait, looking at the image, the variable is $b$, but the expression uses $a$. Let's look closer. Ah, the interior angle is $3b+2$ and the other is $4b+1$. The exterior angle is $5a+10$. This seems to be a typo in the problem itself where it mixes variables $a$ and $b$. However, usually in these sets, if the variable to solve is $b$, the exterior angle likely depends on $b$ or is a constant. Let's re-read carefully. The exterior angle is labeled $5a+10$. The question asks for $b$. There is no information linking $a$ and $b$. Let's assume the exterior angle was meant to be in terms of $b$, perhaps $5b+10$? Or maybe the first interior angle is just a number? No, it says $3b+2$. Let's look at problem 8. It asks for $t$ and has $t$ in the expressions. Problem 7 asks for $b$ but has an $a$ in the exterior angle. This is likely a typo for $5b+10$. Let's solve assuming the exterior angle is $5b+10$.*
* Assumption: Exterior angle is $5b + 10$.
* Remote interior angles: $(3b + 2)^\circ$ and $(4b + 1)^\circ$
* Equation: $(3b + 2) + (4b + 1) = 5b + 10$
* Combine like terms on left: $7b + 3 = 5b + 10$
* Subtract $5b$ from both sides: $2b + 3 = 10$
* Subtract 3 from both sides: $2b = 7$
* Divide by 2: $b = 3.5$

8. Solve for $t$
* Exterior angle: $(50 + 62)^\circ = 112^\circ$
* Remote interior angles: $(33t - 3)^\circ$ and $(45t + 5)^\circ$
* Equation: $(33t - 3) + (45t + 5) = 112$
* Combine like terms: $78t + 2 = 112$
* Subtract 2 from both sides: $78t = 110$
* Divide by 78: $t = \frac{110}{78} = \frac{55}{39} \approx 1.41$

Final Answer:
1. $f \approx 16.86$ ($118/7$)
2. $x = 26$
3. $d = 45$
4. $k \approx -6.33$ ($-19/3$)
5. $m = 4$
6. $g \approx 4.32$ ($82/19$)
7. $b = 3.5$ (Assuming exterior angle is $5b+10$)
8. $t \approx 1.41$ ($55/39$)