1034420finding Missing Angles Worksheet | PDF - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

Problem 1

* Angle $a$: Angles on a straight line add up to $180^\circ$. The angle next to $a$ is $112^\circ$.
$$180 - 112 = 68$$
So, $a = 68^\circ$.
* Angle $b$: Angle $b$ and the $112^\circ$ angle are vertical angles (opposite each other), so they are equal.
So, $b = 112^\circ$.
* Angle $c$: Angle $c$ and angle $a$ are vertical angles, so they are equal.
So, $c = 68^\circ$.

Problem 2

* Angle $a$: All three angles lie on a straight line, so they must add up to $180^\circ$.
$$15 + 38 = 53$$
$$180 - 53 = 127$$
So, $a = 127^\circ$.

Problem 3

* Angle $a$: Angle $a$ and the $40^\circ$ angle are vertical angles.
So, $a = 40^\circ$.
* Angle $b$: Angle $b$ and the $70^\circ$ angle are vertical angles.
So, $b = 70^\circ$.
* Angle $d$: Angles $a$, $d$, and the $40^\circ$ angle form a straight line ($180^\circ$).
$$40 + 40 = 80$$
$$180 - 80 = 100$$
So, $d = 100^\circ$.
* Angle $c$: Angle $c$ and angle $d$ are vertical angles.
So, $c = 100^\circ$.

Problem 4

* Angle $e$: There is a square symbol indicating a right angle ($90^\circ$) between angle $e$ and the $70^\circ$ angle. Together they make a straight line ($180^\circ$).
$$180 - 90 - 70 = 20$$
So, $e = 20^\circ$.
* Angle $d$: Angle $d$ and angle $e$ form a right angle ($90^\circ$) together (based on the perpendicular lines).
$$90 - 20 = 70$$
So, $d = 70^\circ$.
* Angle $c$: Angle $c$ and angle $d$ form a right angle ($90^\circ$) together.
$$90 - 70 = 20$$
So, $c = 20^\circ$.
* Angle $b$: Angle $b$ and the $70^\circ$ angle are vertical angles.
So, $b = 70^\circ$.
* Angle $a$: Angle $a$ and angle $b$ form a right angle ($90^\circ$) together.
$$90 - 70 = 20$$
So, $a = 20^\circ$.

Problem 5

* Angle $a$: Angle $a$ and the $138^\circ$ angle are on a straight line.
$$180 - 138 = 42$$
So, $a = 42^\circ$.
* Angle $b$: Angle $b$ and angle $a$ are vertical angles.
So, $b = 42^\circ$.
* Angle $c$: Angle $c$ and the $132^\circ$ angle are on a straight line.
$$180 - 132 = 48$$
So, $c = 48^\circ$.
* Angle $d$: Angle $d$ and angle $c$ are vertical angles.
So, $d = 48^\circ$.
* Angle $e$: Angles $b$, $e$, and $d$ form a straight line ($180^\circ$).
$$42 + 48 = 90$$
$$180 - 90 = 90$$
So, $e = 90^\circ$.

Problem 6

* Angle $a$: Angle $a$ and the $110^\circ$ angle are on a straight line.
$$180 - 110 = 70$$
So, $a = 70^\circ$.
* Angle $b$: Angle $b$ and the $100^\circ$ angle are on a straight line.
$$180 - 100 = 80$$
So, $b = 80^\circ$.
* Angle $c$: The three angles inside the triangle ($a$, $b$, and $c$) must add up to $180^\circ$.
$$70 + 80 = 150$$
$$180 - 150 = 30$$
So, $c = 30^\circ$.

Problem 7

* Angle $a$: Angle $a$ and the $54^\circ$ angle are alternate interior angles (inside the parallel lines on opposite sides of the transversal). They are equal.
So, $a = 54^\circ$.
* Angle $b$: Angle $b$ and angle $a$ are consecutive interior angles (same side interior). They add up to $180^\circ$.
$$180 - 54 = 126$$
So, $b = 126^\circ$.
* Angle $c$: Angle $c$ and angle $b$ are on a straight line.
$$180 - 126 = 54$$
So, $c = 54^\circ$.

Problem 8

* Angle $a$: This forms a triangle with the bottom left angle ($54^\circ$) and the top angle. First, find the top angle. The top angle and angle $a$ are vertical angles? No, let's look closer.
Let's use the triangle sum. The bottom-left angle inside the triangle corresponds to the $54^\circ$ angle (alternate interior angles). So the bottom-left inside angle is $54^\circ$.
The top angle inside the triangle is vertically opposite to angle $a$? No, angle $a$ is inside the triangle.
Let's re-evaluate based on standard positions.
The angle labeled $54^\circ$ at the bottom is an alternate interior angle to the angle inside the triangle at the bottom left vertex. So, bottom-left internal angle = $54^\circ$.
The angle labeled $a$ is inside the triangle.
Wait, looking at Problem 8 again:
There is a transversal cutting two parallel lines.
Angle $b$ and the $54^\circ$ angle are consecutive interior angles? No, they are on the same side of the transversal but one is exterior and one interior?
Let's look at angle $b$. It is in the position corresponding to the angle supplementary to $54^\circ$ if they were consecutive interior.
Actually, simpler view:
Angle $b$ and the angle adjacent to the $54^\circ$ angle (on the straight line) are alternate interior angles.
Angle adjacent to $54^\circ$ on the straight line = $180 - 54 = 126^\circ$.
Therefore, $b = 126^\circ$ (Alternate Interior Angles).

Now for angle $a$. Angle $a$ is inside a triangle formed by the transversal and another line.
Let's look at the top intersection. Angle $a$ and the angle above it form a linear pair? No.
Let's assume the triangle is formed by the transversal, the top parallel line, and a third line segment.
The angle inside the triangle at the bottom left is $54^\circ$ (Alternate Interior to the given $54^\circ$? No, the given $54^\circ$ is "inside" the parallel lines already? The arrow indicates the line direction.
Let's assume standard "Z" pattern for Alternate Interior Angles.
The $54^\circ$ angle shown is between the parallel lines.
Angle $a$ is also between the parallel lines.
If we look at the triangle containing angle $a$:
The bottom-left angle of that triangle is $54^\circ$ (given).
The top angle of that triangle is vertically opposite to... wait.
Let's look at angle $b$ first. Angle $b$ and the $54^\circ$ angle are consecutive interior angles (same-side interior). They add to $180^\circ$.
$$180 - 54 = 126$$
So, $b = 126^\circ$.

Now for angle $a$. Angle $a$ is part of a triangle. The vertices are:
1. The intersection on the bottom line. The angle inside the triangle here is $54^\circ$.
2. The intersection on the top line.
3. A third point to the right.
This interpretation is tricky without clear labels on the third line.

Let's try a different interpretation which is common for these worksheets:
Angle $a$ and the $54^\circ$ angle are alternate interior angles.
If the line forming angle $a$ is the same transversal as the one forming the $54^\circ$ angle, then $a = 54^\circ$. But there is a triangle drawn.

Let's look at the triangle specifically.
Bottom-left angle = $54^\circ$.
Top angle = $a$.
Right angle = unknown.
However, usually in these problems, if lines look parallel, they are.
Angle $b$ is clearly related to the $54^\circ$ angle. They are consecutive interior angles.
$b + 54 = 180 \rightarrow b = 126^\circ$.

For angle $a$: It appears $a$ and the $54^\circ$ angle are alternate interior angles relative to the transversal on the left.
Therefore, $a = 54^\circ$.

Final Answer:
1. a = 68°, b = 112°, c = 68°
2. a = 127°
3. a = 40°, b = 70°, c = 100°, d = 100°
4. a = 20°, b = 70°, c = 20°, d = 70°, e = 20°
5. a = 42°, b = 42°, c = 48°, d = 48°, e = 90°
6. a = 70°, b = 80°, c = 30°
7. a = 54°, b = 126°, c = 54°
8. a = 54°, b = 126°