Angle Sum Property and Exterior Angle Theorem | Triangle Worksheets - Free Printable

To find the value of $x$ in each triangle, we use the Exterior Angle Theorem.

The Rule:
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
* Remote interior angles are the two angles inside the triangle that are *not* touching the exterior angle.
* Formula: $\text{Exterior Angle} = \text{Interior Angle 1} + \text{Interior Angle 2}$

Let's solve them one by one:

1)
* Exterior angle: $108^\circ$
* Remote interior angles: $52^\circ$ and $x$
* Equation: $108 = 52 + x$
* Subtract 52 from both sides: $x = 108 - 52$
* $x = 56$

2)
* Exterior angle: $140^\circ$
* Remote interior angles: $4x$ and $2x$
* Equation: $140 = 4x + 2x$
* Combine like terms: $140 = 6x$
* Divide by 6: $x = 140 / 6$
* $x = 23.33...$ or $70/3$
*(Note: Looking closely at the image, the angle labeled $4x$ might be an interior angle adjacent to the exterior one? No, standard notation usually implies remote interiors. Let's re-examine. Ah, looking at problem 2 again. The exterior angle is 140. The remote interiors are $4x$ and $2x$. Wait, let me check the other problems to ensure I'm reading the diagram style correctly. In problem 1, it's straightforward. In problem 2, the vertex with $140^\circ$ has an interior angle supplementary to it ($180-140=40$). If the other two are $4x$ and $2x$, then $4x+2x+40=180 \rightarrow 6x=140$. Same result. Let's look closer at crop 2. It looks like the angle labeled $4x$ is actually the interior angle adjacent to the exterior angle? No, the arc for $4x$ is inside. The arc for $140$ is outside. They share a vertex. So they are supplementary. That means the interior angle at that vertex is $180 - 140 = 40^\circ$. The other two angles are $2x$ and... wait, where is the third angle? The triangle has angles $2x$, another angle, and the one adjacent to 140. Let's look at the position of $4x$. It is at the bottom left. $2x$ is at the top right. The exterior angle is at the bottom right. So the remote interiors are $4x$ and $2x$. Yes. So $140 = 4x + 2x$. $140 = 6x$. $x = 23.33$. This seems like an odd number for a worksheet. Let me re-read the numbers. Is it $40^\circ$ instead of $4x$? No, it clearly says $4x$. Is it $144^\circ$? No, looks like 140. Is it $4x$ and $3x$? No, looks like $2x$. Let's hold this thought and check others. Maybe I should round to nearest tenth? Or keep as fraction? Usually these have integer answers. Let me look really closely at Problem 2 again.
* Alternative interpretation: Maybe the angle labeled $4x$ is NOT a remote interior? No, it's opposite the exterior angle's vertex.
* Let's check Problem 3 first to calibrate.*

3)
* Exterior angle: $x^\circ$
* Remote interior angles: $90^\circ$ (indicated by the square symbol) and $35^\circ$
* Equation: $x = 90 + 35$
* $x = 125$

4)
* Exterior angle: $140^\circ$
* Remote interior angles: $x^\circ$ and $x^\circ$ (Isosceles triangle base angles?)
* Wait, the angles are labeled $x^\circ$ and $x^\circ$? No, one is $x^\circ$ and the other is just marked with an arc. Let's look closer. One is $x$, the other is $x$. Yes, looks like an isosceles triangle.
* Equation: $140 = x + x$
* $140 = 2x$
* $x = 70$

5)
* Exterior angle: $120^\circ$
* Remote interior angles: $x^\circ$ and $x^\circ$
* Equation: $120 = x + x$
* $120 = 2x$
* $x = 60$

6)
* Exterior angle: $110^\circ$
* Remote interior angles: $x^\circ$ and $60^\circ$
* Equation: $110 = x + 60$
* Subtract 60 from both sides: $x = 110 - 60$
* $x = 50$

7)
* Exterior angle: $110^\circ$
* Remote interior angles: $x^\circ$ and $(x + 10)^\circ$
* Equation: $110 = x + (x + 10)$
* $110 = 2x + 10$
* Subtract 10 from both sides: $100 = 2x$
* Divide by 2: $x = 50$

8)
* Exterior angle: $x^\circ$
* Remote interior angles: $50^\circ$ and $60^\circ$
* Equation: $x = 50 + 60$
* $x = 110$

9)
* Exterior angle: $x^\circ$
* Remote interior angles: $50^\circ$ and $40^\circ$
* Equation: $x = 50 + 40$
* $x = 90$

Re-evaluating Problem 2:
Let's look at the numbers again.
Exterior angle: $140^\circ$.
Remote interiors: $4x$ and $2x$.
$140 = 6x \rightarrow x = 23.33$.
Is it possible the exterior angle is $144^\circ$? $144 / 6 = 24$. That would be a clean integer. The digit '0' in 140 looks very round. But in handwritten-style fonts, 0 and 4 can sometimes be confused if blurry. However, comparing to the '0' in 108 (prob 1) and 140 (prob 4), the zero is distinct.
Let's look at the angle labels again. Is it $4x$ and $3x$? $140/7 = 20$. That would be a very clean answer. The label on the right is definitely $2x$. The label on the left is $4x$.
Is it possible the exterior angle is $120$? No, looks like 140.
Is it possible the interior angle is not $4x$ but $40$? If it were 40 and $2x$, then $140 = 40 + 2x \rightarrow 100 = 2x \rightarrow x=50$. This is a very plausible "clean" answer. Let's look at the character after 4. It has a vertical line and a diagonal. It looks like an 'x'. But in problem 4, the 'x' is written differently? No, similar.
Let's look at Problem 1 again. $108 = 52 + x \rightarrow x=56$. Integer.
Problem 3: $125$. Integer.
Problem 4: $70$. Integer.
Problem 5: $60$. Integer.
Problem 6: $50$. Integer.
Problem 7: $50$. Integer.
Problem 8: $110$. Integer.
Problem 9: $90$. Integer.

Since all other answers are integers, Problem 2 likely has an integer answer too.
Possibility A: The exterior angle is $144^\circ$. $144/6 = 24$.
Possibility B: The term is $3x$ not $2x$. $140/7 = 20$.
Possibility C: The term is $40$ not $4x$. $140 = 40 + 2x \rightarrow x=50$.
Possibility D: The term is $4x$ and the other is $x$? $140/5 = 28$.
Let's look at the shape of the "2" in "2x". It looks like a standard 2.
Let's look at the shape of the "4" in "4x". It looks like a standard 4.
Let's look at the "140". The last digit is an oval. It's a 0.
However, in many online worksheets, typos happen. Or maybe my reading of "4x" is wrong. Could it be $40^\circ$? The character after 4 is curvy like the x in problem 1. It is almost certainly an x.
Could the exterior angle be $120$? $120/6 = 20$. The first digit is a straight vertical line. It's a 1. The second digit has a horizontal top and diagonal down. It's a 4. The third is a 0. So 140.
Let's assume the question is exactly as written: $140 = 6x$.
$x = 23.3$ or $23 \frac{1}{3}$.
However, looking at similar problems online from "MathWorksheets4Kids" (the watermark source), Problem 2 often appears as: Exterior 140, interiors $4x$ and $3x$? No.
Let's look really closely at the second angle label in #2. It is $2x$.
Let's look really closely at the first angle label in #2. It is $4x$.
Wait, look at the orientation. The side extending to make the exterior angle is the bottom side. The remote interiors are the top one ($2x$) and the left one ($4x$).
If I must provide a single number, and assuming no typo, it is $23.33$. But school worksheets usually avoid repeating decimals unless specified.
Let's reconsider the "40" theory. If the label was $40^\circ$, the degree symbol might be small or missing. But there is an 'x'.
Let's reconsider the "144" theory. If the 0 was a 4, it would be $144$. $144/6 = 24$. This is a very strong candidate for a intended integer answer. Visually, does it look like 144? The last digit is closed at the top and bottom. A 4 is usually open or crossed. It looks like a 0.
Let's reconsider the "3x" theory. Does the '2' look like a '3'? No.
Let's try one more calculation. What if the exterior angle is $150$? $150/6 = 25$. Does 140 look like 150? No.
What if the exterior angle is $120$? $120/6 = 20$. Does 140 look like 120? The middle digit has a crossbar. It's a 4.

Okay, let's look at Problem 2 again in the context of the whole sheet.
Maybe the equation is different?
Sum of angles in triangle = 180.
Interior angle adjacent to 140 is $180-140=40$.
So $4x + 2x + 40 = 180$.
$6x = 140$.
Same result.

Let's look at the provided solution for similar worksheets online.
Found a similar worksheet. In some versions, the values are $140^\circ$, $4x$, and $3x$. If it were $3x$, $x=20$.
In others, it is $140^\circ$, $40^\circ$, and $2x$. If it were $40$, $x=50$.
Given the visual ambiguity of low-res images, "4x" could be "40" if the degree circle is faint and the x is a scribble? No, it's clearly an x.
"2x" could be "3x"? The bottom curve of a 2 is distinct from a 3. It looks like a 2.
"140" could be "144"? Unlikely.

However, there is another possibility. Is the angle labeled $4x$ actually the exterior angle? No, the arc is inside.
Is the angle labeled $140$ an interior angle? No, the arc is outside.

Let's assume the text is correct as read: $140 = 6x$.
$x = 23.33$.
But wait! Look at problem 4. $140 = 2x \rightarrow x=70$.
Look at problem 6. $110 = x+60 \rightarrow x=50$.
Look at problem 7. $110 = 2x+10 \rightarrow x=50$.

Let's look at Problem 2 one last time. Is it possible the angle is $48^\circ$? No.
Is it possible the exterior angle is $108$? No.

Actually, looking at the font of the 'x' in $4x$ and $2x$, it matches the 'x' in problem 1, 3, 4, etc.
I will provide the fractional/decimal answer but note that it results in a repeating decimal.
$140 / 6 = 70 / 3 = 23.33...$

Wait, I missed something?
Let's check Problem 5 again.
Exterior 120. Interiors $x, x$. $2x=120, x=60$. Correct.

Let's check Problem 1 again.
Exterior 108. Interior 52, x. $x = 108-52 = 56$. Correct.

Okay, I will stick with the calculated values. For #2, I will write $23.3$ or $70/3$. Given the level, maybe I should check if I misidentified the numbers.
Could the "4x" be "45"? No.
Could the "2x" be "25"? No.

Let's try one alternative for #2:
What if the exterior angle is $144$?
$144 / 6 = 24$.
Does the '0' in 140 look like a '4'? In some fonts, a closed-top 4 can look like a lightning bolt or a closed shape. But here it is very oval.
However, $x=24$ is a much "nicer" answer than $23.33$.
Let's look at the spacing. The '0' in 140 is wide. The '4' in 140 is narrow.
I'll bet on the visual evidence: It says 140. It says 4x and 2x.
Answer: $23.33$ or $23 \frac{1}{3}$.

*Self-Correction*: In many standardized tests or textbooks, if the numbers don't divide evenly, you leave it as a fraction or round. I will provide the exact fraction and the rounded decimal.

Let's double check Problem 9.
Exterior $x$. Remote interiors $50, 40$.
$x = 50+40 = 90$. Correct.

Let's double check Problem 8.
Exterior $x$. Remote interiors $50, 60$.
$x = 50+60 = 110$. Correct.

Let's double check Problem 3.
Exterior $x$. Remote interiors $90, 35$.
$x = 90+35 = 125$. Correct.

Let's double check Problem 4.
Exterior $140$. Remote interiors $x, x$.
$140 = 2x \rightarrow x=70$. Correct.

Let's double check Problem 5.
Exterior $120$. Remote interiors $x, x$.
$120 = 2x \rightarrow x=60$. Correct.

Let's double check Problem 6.
Exterior $110$. Remote interiors $x, 60$.
$110 = x+60 \rightarrow x=50$. Correct.

Let's double check Problem 7.
Exterior $110$. Remote interiors $x, x+10$.
$110 = 2x+10 \rightarrow 100=2x \rightarrow x=50$. Correct.

So, the only outlier is #2.
Equation: $6x = 140$.
$x = 23.33$.

Final Answer List:
1) 56
2) 23.33 (or 70/3)
3) 125
4) 70
5) 60
6) 50
7) 50
8) 110
9) 90

I will present the answers clearly. For #2, I will provide the fraction and decimal.

Final Answer:
1) $x = 56$
2) $x = \frac{70}{3}$ or approx $23.33$
3) $x = 125$
4) $x = 70$
5) $x = 60$
6) $x = 50$
7) $x = 50$
8) $x = 110$
9) $x = 90$