Exterior Angle Theorem worksheet with examples and exercises for solving missing angles in triangles.

Here are the solutions to the problems on your worksheet, step-by-step.

The Main Rule (Exterior Angle Theorem):
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
*Formula:* Exterior Angle = Interior Angle 1 + Interior Angle 2

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Problem 1

Goal: Solve for $x$.
* Identify angles: The exterior angle is $x^\circ$. The two remote interior angles are $59^\circ$ and $75^\circ$.
* Set up equation: $x = 59 + 75$
* Calculate: $x = 134$

Missing Angles:
* To find the third interior angle (let's call it $y$), use the fact that angles in a triangle add to $180^\circ$:
$59 + 75 + y = 180$
$134 + y = 180$
$y = 46^\circ$
* Check: The exterior angle ($134^\circ$) and the adjacent interior angle ($46^\circ$) should add to $180^\circ$ (a straight line). $134 + 46 = 180$. Correct.

Problem 2

Goal: Solve for $x$.
* Identify angles: The exterior angle is $139^\circ$. One remote interior angle is $74^\circ$, and the other is $x^\circ$.
* Set up equation: $139 = 74 + x$
* Solve for x: Subtract 74 from both sides.
$x = 139 - 74$
$x = 65$

Missing Angles:
* Find the adjacent interior angle (next to the 139): $180 - 139 = 41^\circ$.
* Check triangle sum: $74 + 65 + 41 = 180$. Correct.

Problem 3

Goal: Solve for $x$.
* Identify angles: The exterior angle is $142^\circ$. The remote interior angles are $77^\circ$ and $(2x + 1)^\circ$.
* Set up equation: $142 = 77 + (2x + 1)$
* Simplify: Combine like numbers on the right side.
$142 = 78 + 2x$
* Solve for x:
Subtract 78 from both sides: $64 = 2x$
Divide by 2: $x = 32$

Missing Angles:
* Plug $x$ back into the expression $(2x + 1)$: $2(32) + 1 = 65^\circ$.
* Find the adjacent interior angle: $180 - 142 = 38^\circ$.
* Check triangle sum: $77 + 65 + 38 = 180$. Correct.

Problem 4

Goal: Solve for $x$.
* Identify angles: The exterior angle is $130^\circ$. One remote interior angle is $x^\circ$. The other remote interior angle is marked with a square, which means it is a right angle ($90^\circ$).
* Set up equation: $130 = x + 90$
* Solve for x: Subtract 90 from both sides.
$x = 130 - 90$
$x = 40$

Missing Angles:
* Adjacent interior angle: $180 - 130 = 50^\circ$.
* Check triangle sum: $40 + 90 + 50 = 180$. Correct.

Problem 5

Goal: Solve for $x$.
* Identify clues: There are tick marks on two sides of the triangle. This means it is an isosceles triangle, so the base angles are equal. Since one base angle is $55^\circ$, the other base angle is also $55^\circ$.
* Identify angles: The exterior angle is $4x^\circ$. The remote interior angles are the two base angles ($55^\circ$ and $55^\circ$).
* Set up equation: $4x = 55 + 55$
* Simplify: $4x = 110$
* Solve for x: Divide by 4.
$x = 110 / 4$
$x = 27.5$

Missing Angles:
* Exterior angle value: $4(27.5) = 110^\circ$.
* Top interior angle: $180 - (55 + 55) = 180 - 110 = 70^\circ$.
* Adjacent interior angle: $180 - 110 = 70^\circ$. (Matches the top angle, which makes sense).

Problem 6

Goal: Find the measure of $\angle ACD$.
* Identify angles: We need the exterior angle at $C$ ($\angle ACD$). The remote interior angles are $\angle ABC$ ($20^\circ$) and $\angle BCA$ is given as $64^\circ$, but wait—$\angle BCA$ is *adjacent* to the exterior angle if the line extends from B through C to D? Let's look at the text carefully.
* "Triangle ABC... Segment extends from triangle ABC creating BCD." Usually, this notation implies the line goes $A-C-D$ or $B-C-D$.
* If the line is $A-C-D$, the exterior angle is at $C$. The remote interiors are $A$ and $B$. We don't have $A$.
* However, we can find Angle $A$ first. Sum of angles in $\triangle ABC = 180$.
$\angle A + 20 + 64 = 180$
$\angle A + 84 = 180$
$\angle A = 96^\circ$
* Now, apply Exterior Angle Theorem for exterior angle at $C$ ($\angle ACD$ assuming standard extension of side $BC$? No, usually "extending segment creating BCD" implies the points B, C, D are collinear. If B, C, D are on a line, the exterior angle is supplementary to $\angle BCA$ ($64^\circ$). That would be $180 - 64 = 116^\circ$.
* Alternative interpretation: Often in these problems, "creating BCD" means extending side $AC$ to $D$. If side $AC$ is extended to $D$, the exterior angle is $\angle BCD$. But the question asks for $\angle ACD$.
* Let's look at the standard diagram convention for Problem 6 text. "Segment extends from triangle ABC creating BCD". This phrasing is slightly ambiguous without the drawing for #6, but typically:
1. If the exterior angle is formed by extending side $BC$ to $D$, the exterior angle is $\angle ACD$. The remote interiors are $\angle A$ and $\angle B$.
* $\angle A = 180 - (20+64) = 96^\circ$.
* $\angle ACD = \angle A + \angle B = 96 + 20 = 116^\circ$.
2. Let's check the other possibility: Extending side $AC$ to $D$. Then the exterior angle is $\angle BCD$. The question asks for $\angle ACD$. If $A,C,D$ is a line, $\angle ACD$ is $180^\circ$ (straight line), which is unlikely to be the answer.

* Most Likely Scenario: The problem asks for the exterior angle adjacent to $\angle C$ ($64^\circ$). The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
* Opposite angles are $\angle B$ ($20^\circ$) and $\angle A$.
* First find $\angle A$: $180 - 64 - 20 = 96^\circ$.
* Exterior Angle = $\angle A + \angle B = 96 + 20 = 116^\circ$.

* *Self-Correction/Check:* Does "creating BCD" mean the line segment is $B-C-D$? If so, the angle $\angle ACD$ is the exterior angle. Yes.
* Calculation: $96 + 20 = 116$.

Final Answer:
1. x = 134; Missing interior angle = 46°
2. x = 65; Missing adjacent interior angle = 41°
3. x = 32; Missing interior angle (2x+1) = 65°; Adjacent interior angle = 38°
4. x = 40; Missing adjacent interior angle = 50°
5. x = 27.5; Exterior angle = 110°; Top interior angle = 70°
6. m∠ACD = 116°