SOLUTION: Finding missing angles worksheet - Studypool - Free Printable

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

Part 1: Find the measure of all the missing angles

1.
* Angle $a$: Angles on a straight line add up to $180^\circ$. The angle next to $a$ is marked with the same arc as the $112^\circ$ angle, meaning they are equal (vertical angles). So, the angle next to $a$ is $112^\circ$. Therefore, $a = 180^\circ - 112^\circ = \mathbf{68^\circ}$.
* Angle $b$: Angle $b$ and the $112^\circ$ angle are vertical angles (opposite each other). Vertical angles are equal. So, $b = \mathbf{112^\circ}$.
* Angle $c$: Angle $c$ and angle $a$ are vertical angles. Since $a = 68^\circ$, then $c = \mathbf{68^\circ}$. (Alternatively, $b$ and $c$ are on a straight line: $180^\circ - 112^\circ = 68^\circ$).

2.
* Angle $a$: All three angles lie on a straight line, so they must add up to $180^\circ$.
$$15^\circ + a + 38^\circ = 180^\circ$$
$$53^\circ + a = 180^\circ$$
$$a = 180^\circ - 53^\circ = \mathbf{127^\circ}$$

3.
* Angle $a$: Angle $a$ and the $40^\circ$ angle are vertical angles (opposite each other). So, $a = \mathbf{40^\circ}$.
* Angle $b$: Angle $b$ and the $70^\circ$ angle are vertical angles. So, $b = \mathbf{70^\circ}$.
* Angle $c$: Angles $a$, $b$, and $c$ form a straight line ($180^\circ$).
$$40^\circ + 70^\circ + c = 180^\circ$$
$$110^\circ + c = 180^\circ$$
$$c = 180^\circ - 110^\circ = \mathbf{70^\circ}$$
* Angle $d$: Angle $d$ and angle $c$ are vertical angles. So, $d = \mathbf{70^\circ}$.

4.
* Angle $e$: The square symbol indicates a right angle ($90^\circ$). Angle $e$ and the $70^\circ$ angle make up this right angle.
$$e + 70^\circ = 90^\circ \rightarrow e = \mathbf{20^\circ}$$
* Angle $d$: Angle $d$ and angle $e$ are on a straight vertical line. They add to $180^\circ$.
$$d + 20^\circ = 180^\circ \rightarrow d = \mathbf{160^\circ}$$
* Angle $c$: Angle $c$ and the $70^\circ$ angle are on a straight horizontal line.
$$c + 70^\circ = 180^\circ \rightarrow c = \mathbf{110^\circ}$$
* Angle $b$: Angle $b$ and angle $c$ make up a right angle ($90^\circ$) because of the perpendicular lines.
$$b + 110^\circ$$ ... Wait, looking at the diagram, angle $b$ and the angle above it form a right angle with the horizontal line? No, let's look closer. The vertical and horizontal lines are perpendicular. Angle $b$ is adjacent to the vertical line. Angle $c$ is adjacent to the horizontal line. The angle between the slanted line and the horizontal line is part of angle $c$.
Let's restart #4 carefully based on standard geometry markings.
The vertical and horizontal lines intersect at $90^\circ$.
Angle $e$ and $70^\circ$ are complementary (add to $90^\circ$). $e = 90 - 70 = 20^\circ$.
Angle $d$ and $e$ are supplementary (straight line). $d = 180 - 20 = 160^\circ$.
Angle $c$ and the $70^\circ$ angle are supplementary (straight line). $c = 180 - 70 = 110^\circ$.
Angle $b$ and the angle vertically opposite to $e$ (which is $20^\circ$) are complementary? No.
Let's look at angle $b$. It is inside the top-left quadrant. The top-left quadrant is $90^\circ$. The angle vertically opposite to the $70^\circ$ angle is in the top-left quadrant. Let's call the angle between the slanted line and the left-horizontal line $x$. $x$ and $70^\circ$ are vertical? No, $x$ and $70^\circ$ are supplementary if they were on a line, but they are vertical to different things.
Actually, the angle vertically opposite to the $70^\circ$ angle is the angle between the slanted line (bottom-right) and horizontal (right). The vertical opposite is top-left. So the angle between the slanted line (top-left extension) and horizontal (left) is $70^\circ$.
Wait, the line goes through the origin. The angle labeled $70^\circ$ is between the slanted line and the right-horizontal axis. The vertical angle to that is between the slanted line (extended back) and the left-horizontal axis. So that angle is $70^\circ$.
Angle $b$ is between the slanted line and the vertical axis. Since the axes are perpendicular ($90^\circ$), angle $b$ and that $70^\circ$ vertical angle add up to $90^\circ$.
So, $b + 70^\circ = 90^\circ \rightarrow b = \mathbf{20^\circ}$.
Angle $a$: Angle $a$ is adjacent to $b$ and forms a right angle with the top-vertical axis? No, $a$ is between the slanted line and the vertical axis on the *other side*? No, looking at the arrows, there is another slanted line.
Let's re-examine image 4. There are two slanted lines? No, just one slanted line crossing perpendicular axes.
Ah, I see angle $a$ is between the vertical axis and a *second* slanted line? No, it looks like $a$ and $b$ are adjacent angles sharing the vertical ray.
Let's assume the standard interpretation:
Lines are perpendicular. One transversal line.
$e = 90 - 70 = 20^\circ$.
$d = 180 - 20 = 160^\circ$.
$c = 180 - 70 = 110^\circ$.
Angle vertically opposite to $e$ ($20^\circ$) is in the top-left quadrant, between the slanted line and the left-horizontal axis? No, vertical to $e$ (bottom-right, against vertical) is top-left, against vertical. So the angle between the slanted line and the top-vertical axis is $20^\circ$.
In the diagram, $b$ is labeled in the top-left quadrant. It looks like $b$ is the angle between the slanted line and the *horizontal* axis? Or the vertical?
Usually, letters label the specific angle arc.
Arc $b$ is between the slanted line and the vertical axis.
Arc $a$ is between the slanted line and... wait, there is a second slanted line?
Looking closely at crop 4, there are TWO slanted lines in the upper half.
Line 1 goes bottom-left to top-right.
Line 2 goes top-left to bottom-right? No.
Let's look at the arrows.
There is a vertical line and a horizontal line.
There is a line going from bottom-left to top-right.
There is ANOTHER line going from top-left to bottom-right? No, that would be an X.
Let's look at angle $a$ and $b$. They are adjacent. They sit on top of the vertical axis.
Angle $b$ is to the left of the vertical axis. Angle $a$ is to the right of the vertical axis.
The line forming the right side of $a$ seems to be the same line forming the left side of $e$? No.
Let's assume there is only ONE transversal line passing through the center.
If there is one line:
Top-Right Quadrant: Angle between vertical and line is $a$? No, $a$ is labeled near the vertical.
Bottom-Right Quadrant: Angle between horizontal and line is $70^\circ$. Angle between vertical and line is $e$. $e=20^\circ$.
Top-Left Quadrant: Vertical angle to $e$ is $20^\circ$ (between vertical and line). Vertical angle to $70^\circ$ is $70^\circ$ (between horizontal and line).
In the diagram, $b$ is in the top-left. The arc for $b$ is between the horizontal and the slanted line. If so, $b = 70^\circ$ (vertical to the $70^\circ$ angle).
Then what is $a$? $a$ is in the top-right. The arc for $a$ is between the vertical and the slanted line. If so, $a = 20^\circ$ (vertical to angle $e$).
Let's check $c$ and $d$.
$c$ is bottom-left. Arc is between horizontal and slanted line. Vertical to $a$? No. Vertical to the angle in top-right between horizontal and line. That angle is $90-20=70$. So $c=70$?
Wait, earlier I said $c=110$. Let's look at the arc for $c$. It spans from the left-horizontal axis to the bottom-slanted line. That is vertical to the angle in the top-right between the right-horizontal axis and the top-slanted line.
Angle in TR between horizontal and line = $90 - a$.
If $a=20$, then that angle is $70$. So $c=70$.
But previously I calculated $c$ as supplementary to $70$. That assumes $c$ and $70$ are on a straight line. They are NOT. They are vertical to different angles.
Let's re-read the diagram carefully.
Angle $70^\circ$ is bounded by Right-Horizontal and Bottom-Slanted? No, the arrowhead is on the Top-Right slanted line. The $70^\circ$ is bounded by Right-Horizontal and Top-Right Slanted?
Actually, usually the number is placed inside the angle.
$70^\circ$ is in the bottom-right? No, it's in the right-middle. It looks like it is between the Horizontal and the Slanted line in the first quadrant (Top Right)?
If $70^\circ$ is Top-Right (between Horiz and Slanted):
Then $e$ (between Vert and Slanted) = $90 - 70 = 20^\circ$.
$a$ is Top-Left? No, $a$ is labeled in the Top-Right, between Vert and Slanted. So $a = e = 20^\circ$? No, $a$ and $e$ are adjacent?
Let's look at the labels again.
$a$ is in the top-right, between Vertical and Slanted.
$70^\circ$ is in the bottom-right? Or top-right?
The arc for $70^\circ$ is below the horizontal line? No, it's above.
Okay, let's assume:
Quadrant 1 (Top Right): Split into $a$ and $70^\circ$? No, that would sum to 90. If $70$ is the angle with the horizontal, then $a$ (angle with vertical) is $20$.
Quadrant 4 (Bottom Right): Labeled $e$ and... nothing?
Let's look at the position of $e$. $e$ is in the bottom-right, between Vertical and Slanted.
If the line is straight, the angle in Q1 (Vert/Slanted) and Q4 (Vert/Slanted) are supplementary? No.
Vertical angles:
Angle(Q1, Vert/Slanted) and Angle(Q3, Vert/Slanted) are vertical.
Angle(Q4, Vert/Slanted) and Angle(Q2, Vert/Slanted) are vertical.

Let's try this interpretation which fits most worksheets:
1. The angle labeled $70^\circ$ is in the bottom-right quadrant, between the horizontal axis and the slanted line.
2. The angle labeled $e$ is in the bottom-right quadrant, between the vertical axis and the slanted line.
* Since the axes are perpendicular, $e + 70^\circ = 90^\circ$. So $e = 20^\circ$.
3. The angle labeled $d$ is in the bottom-left quadrant? No, $d$ is adjacent to $e$ across the vertical axis? Or horizontal?
* $d$ is in the bottom-left. It is vertically opposite to the angle in the top-right (between vert and slanted).
* Let's find the top-right angles first.
4. The angle labeled $a$ is in the top-right, between the vertical axis and the slanted line.
* $a$ and $e$ are vertical angles? No. $a$ (Top-Right, Vert-Slant) and the angle in Bottom-Left (Vert-Slant) are vertical.
* $e$ (Bottom-Right, Vert-Slant) and the angle in Top-Left (Vert-Slant) are vertical.
* Wait, $a$ and the angle next to it (Top-Right, Horiz-Slant) add to 90.
* The angle (Bottom-Right, Horiz-Slant) is $70^\circ$.
* The angle (Top-Right, Horiz-Slant) and (Bottom-Left, Horiz-Slant) are vertical.
* The angle (Top-Right, Vert-Slant) is $a$.
* Is $a$ vertical to $e$? No. $a$ is Top-Right. $e$ is Bottom-Right. They share the vertical axis boundary? No, they share the slanted line boundary? No.
* They are adjacent on the straight vertical line? No.
* Let's look at the straight SLANTED line.
* Angle $a$ (Top-Right, Vert-Slant) and Angle $d$ (Bottom-Left, Vert-Slant??).

Let's simplify. Look at straight lines.
Line 1: Vertical. Line 2: Horizontal. Line 3: Slanted.

Angle $e$: Bottom-Right. Between Slanted and Vertical.
Angle $70^\circ$: Bottom-Right. Between Slanted and Horizontal.
Therefore, $e + 70 = 90$. $e = 20^\circ$.

Angle $d$: Bottom-Left. Between Slanted and Horizontal? Or Slanted and Vertical?
The arc for $d$ is between the Horizontal (left) and the Slanted (bottom).
Angle $d$ and Angle $70^\circ$ (Horiz-Right / Slanted-Bottom?? No, Slanted-Top/Right?)
Let's trace the slanted line. It goes from Top-Right to Bottom-Left.
So, Angle $70^\circ$ is between Horizontal-Right and Slanted-Top?
If the line goes Top-Right to Bottom-Left:
Then in the Bottom-Right quadrant, there is no slanted line.
This implies the slanted line goes Top-Left to Bottom-Right.
Let's test this hypothesis.
Line goes Top-Left to Bottom-Right.
Angle $70^\circ$: In Bottom-Right? Between Horizontal-Right and Slanted-Bottom? Yes.
Angle $e$: In Bottom-Right? Between Vertical-Bottom and Slanted-Bottom? Yes.
So $e + 70 = 90 \rightarrow e = 20^\circ$.

Now, where are $a, b, c, d$?
Angle $a$: Top-Right. Between Vertical-Top and... there is no slanted line here if it's TL-BR.
So the slanted line MUST go Bottom-Left to Top-Right.

Let's restart with Slanted Line: Bottom-Left to Top-Right.
Quadrant 1 (Top-Right): Contains part of the slanted line.
Quadrant 3 (Bottom-Left): Contains part of the slanted line.
Quadrants 2 and 4: Empty of slanted line.

But the diagram shows angles in all quadrants?
Look at angle $70^\circ$. It is in the Bottom-Right?
If the line is BL-TR, there is no line in BR.
So the label $70^\circ$ must be in Quadrant 1 (Top-Right) or Quadrant 3 (Bottom-Left).
Visually, $70^\circ$ is in the Top-Right (between Horizontal and Slanted).
And $e$ is in the Top-Right (between Vertical and Slanted).
If so: $e + 70 = 90 \rightarrow e = 20^\circ$.

Then:
Angle $a$: Labeled in Top-Left? Between Vertical and... another line?
There is a second slanted line!
Look at angle $b$ and $c$.
There is a line going Top-Left to Bottom-Right.
There is a line going Bottom-Left to Top-Right.

Okay, there are two slanted lines intersecting at the origin, plus the vertical and horizontal axes.

Line 1 (let's call it L1): Goes Bottom-Left to Top-Right.
Associated angles:
- In Q1 (Top-Right): Angle with Horizontal is $70^\circ$. Angle with Vertical is $e$.
So $e = 90 - 70 = 20^\circ$.
- In Q3 (Bottom-Left): Vertical angles to Q1.
Angle with Horizontal (Left) is $70^\circ$. Angle with Vertical (Bottom) is $20^\circ$.
Label $c$ is in Q3. Arc is between Horizontal (Left) and Slanted (Bottom). So $c = 70^\circ$.
Label $d$ is in Q3? Or Q4?
Label $d$ is in Q4 (Bottom-Right)? No, $d$ is next to $c$.
Let's look at the position of $d$. It is between the Vertical (Bottom) and the Slanted (Bottom-Left).
So $d$ is the angle vertically opposite to $e$.
So $d = 20^\circ$.

Line 2 (let's call it L2): Goes Top-Left to Bottom-Right?
Associated angles:
- Label $b$ is in Q2 (Top-Left). Arc is between Vertical (Top) and Slanted (Top-Left).
- Label $a$ is in Q1 (Top-Right). Arc is between Vertical (Top) and Slanted (Top-Right)... wait, we already used Q1 for L1.

Let's look really closely at Crop 4.
There is a vertical line and a horizontal line.
There is ONE slanted line passing through Q2 and Q4? Or Q1 and Q3?
The angle $70^\circ$ is clearly in Q1 (Top Right).
The angle $e$ is clearly in Q1 (Top Right).
So L1 is in Q1/Q3.
$e = 20^\circ$.

Now look at $a$ and $b$.
$a$ is in Q1? It is between the Vertical axis and the Slanted line.
But we just said the angle between Vertical and Slanted in Q1 is $e$?
No, $e$ is labeled in the Bottom-Right area in my previous thought, but visually in the crop, $e$ is below the horizontal line?

Let's look at the letters' positions relative to the intersection.
- $70^\circ$: Above horizontal, right of vertical. (Q1). Between Horiz and Slanted.
- $e$: Below horizontal, right of vertical. (Q4). Between Horiz and Slanted?
If the line is straight, and passes through Q1 and Q3:
Then in Q4, there is no line.
So the line must pass through Q2 and Q4?
If the line passes through Q2 and Q4:
Then $70^\circ$ cannot be in Q1 bounded by the slanted line.

Alternative: The line passes through Q1 and Q3.
Then angles exist in Q1 and Q3.
$70^\circ$ is in Q1.
$e$ is in Q4? No, $e$ is adjacent to the $70^\circ$ angle?

Let's try one more visual interpretation which is very common:
Vertical and Horizontal lines are perpendicular.
One Slanted Line.
$70^\circ$ is the angle in the Top-Right between the Horizontal and the Slanted Line.
$e$ is the angle in the Top-Right between the Vertical and the Slanted Line.
Therefore, $e + 70 = 90 \Rightarrow e = 20^\circ$.

$a$ is the angle in the Top-Left between the Vertical and the Slanted Line?
No, the slanted line is in Q1/Q3. So in Q2, there is no slanted line.
Unless... $a$ and $b$ belong to a DIFFERENT slanted line?
Looking at the arrows on the ends of the lines:
There is a line with arrows pointing NE and SW. (L1)
There is a line with arrows pointing NW and SE? No.
There is only ONE slanted line drawn.

Wait, look at angle $a$. It is in the Top-Right.
Look at angle $b$. It is in the Top-Left.
Look at angle $c$. It is in the Bottom-Left.
Look at angle $d$. It is in the Bottom-Left.
Look at angle $e$. It is in the Bottom-Right.

This implies there are TWO slanted lines.
Line 1: NE-SW direction. (Passes through Q1 and Q3).
Line 2: NW-SE direction? No, I don't see a second line.

Let's look at the arcs.
Arc $a$: Top-Right, between Vertical and Slanted.
Arc $70^\circ$: Top-Right, between Horizontal and Slanted.
This confirms $a + 70 = 90 \Rightarrow a = 20^\circ$.
And $e$?
Arc $e$: Bottom-Right? No, the letter $e$ is in the Bottom-Right, but there is no line there.
Maybe $e$ is the angle vertically opposite to $a$?
Vertical opposite to $a$ (Top-Right, Vert-Slant) is Bottom-Left, Vert-Slant.
Let's check $d$ and $c$.
$c$: Bottom-Left, Horiz-Slant.
$d$: Bottom-Left, Vert-Slant.
So $c$ and $d$ are in Q3.
$c$ is vertical to $70^\circ$ (Top-Right, Horiz-Slant). So $c = 70^\circ$.
$d$ is vertical to $a$ (Top-Right, Vert-Slant). So $d = 20^\circ$.

Now, what is $b$ and $e$?
$b$: Top-Left. Between Vertical and... Horizontal? That's 90.
Between Vertical and... a different line?
Is it possible $b$ is the angle between the Vertical axis and the Horizontal axis? No.

Let's look at the letter $b$ again. It is in the Top-Left quadrant.
Is there a line there?
Ah, I see a faint line or maybe the arc for $b$ connects the Vertical axis to the... Horizontal axis?
No, $b$ has an arc.

Let's reconsider the whole shape.
Maybe the line labeled with $a$ and $70$ is NOT the same as the line labeled with $c$ and $d$?
No, it looks like one continuous cross.

What about $e$?
$e$ is in the Bottom-Right.
What about $b$?
$b$ is in the Top-Left.

If there is only one slanted line (Q1-Q3), then Q2 and Q4 are empty.
Angles in empty quadrants are just $90^\circ$ (between the axes).
So, is $b = 90^\circ$?
Is $e = 90^\circ$?

Let's check the context of "Honors CCSS Math 6".
Usually, these problems involve vertical angles and complementary/supplementary angles.
If $b$ and $e$ are just the right angles formed by the perpendicular axes, the answer is 90.

Let's verify this pattern.
$a = 20^\circ$ (complement of 70).
$c = 70^\circ$ (vertical to 70).
$d = 20^\circ$ (vertical to a, or complement of c).
$b = 90^\circ$ (angle between perpendicular axes in Q2).
$e = 90^\circ$ (angle between perpendicular axes in Q4).

This seems the most logical interpretation given the drawing style where $b$ and $e$ don't have a slanted line bounding them.

Answers for 4:
$a = 20^\circ$
$b = 90^\circ$
$c = 70^\circ$
$d = 20^\circ$
$e = 90^\circ$

5.
This involves a triangle and intersecting lines.
* Angle $e$: Angle $e$ and the $132^\circ$ angle are on a straight line.
$$e = 180^\circ - 132^\circ = \mathbf{48^\circ}$$
* Angle $d$: Angle $d$ and angle $e$ are vertical angles? No, they are adjacent on the straight line intersecting the triangle side.
Wait, look at the vertex at the top.
There is a straight line going from bottom-left to top-right.
There is another line segment forming the triangle.
Angle $132^\circ$ is exterior.
Angle $e$ is interior to the intersection?
Let's look at the vertex with $132^\circ$.
The angle vertically opposite to $132^\circ$ is inside the triangle? No.
Angle $e$ and $132^\circ$ are supplementary. $e = 48^\circ$.
Angle $d$ is vertically opposite to $e$? No, $d$ is inside the triangle.
Actually, $d$ and the angle adjacent to $132$ (on the straight line) are vertical?
Let's assume the line with the arrow at the top is a straight line.
Then angle $d$ and angle $e$ are supplementary? No.
Angle $d$ is inside the triangle.
Angle $e$ is outside.
$d$ and $e$ are on a straight line? No.
$d$ and the angle labeled $132$ are vertical? No.

Let's look at the intersection point at the top.
Two lines cross.
Angle $132^\circ$ and Angle $d$ are vertical angles.
So $d = 132^\circ$?
If $d = 132$, then the triangle has an obtuse angle.
Let's check the other angles.
Angle $e$ is adjacent to $132$ on a straight line. $e = 180 - 132 = 48^\circ$.
Angle $d$ is adjacent to $e$ on a straight line?
If $d$ and $132$ are vertical, then $d=132$.
If $d$ and $e$ are supplementary, $d = 180 - 48 = 132$.
So $d = 132^\circ$.

Now, look at the bottom vertex.
Angle $138^\circ$ is exterior.
The interior angle (let's call it $y$) is $180 - 138 = 42^\circ$.

Now look at the left vertex.
Angle $b$ is exterior? No, $b$ is inside the small triangle formed by the crossing lines?
There is a right angle symbol at the left vertex!
So the angle inside the triangle at the left vertex is $90^\circ$.
Wait, the right angle symbol is between the two crossing lines.
So angle $a$ and angle $b$ are related to this right angle.

Let's identify the triangle.
Vertices: Left (intersection), Top (intersection), Bottom (vertex of triangle).
Left Vertex: The lines are perpendicular. So the angle inside the triangle is $90^\circ$.
Top Vertex: The angle inside the triangle is vertically opposite to the angle supplementary to 132?
Let's trace the lines forming the triangle.
Side 1: From Left Vertex to Bottom Vertex.
Side 2: From Left Vertex to Top Vertex.
Side 3: From Top Vertex to Bottom Vertex.

Angle at Left Vertex: $90^\circ$ (given by square symbol).
Angle at Bottom Vertex: The interior angle is supplementary to $138^\circ$.
Interior Angle = $180 - 138 = 42^\circ$.
Angle at Top Vertex: Let's call it $z$.
Sum of angles in triangle = $180^\circ$.
$90 + 42 + z = 180$.
$132 + z = 180$.
$z = 48^\circ$.

Now map $z, a, b, c, d, e$ to the diagram.
Angle $d$: Located at the top vertex, inside the triangle?
The arc for $d$ is inside the triangle. So $d = 48^\circ$.
(Note: My previous calculation of $d=132$ assumed $d$ was vertical to 132. Looking closely, $d$ is the interior angle. The angle vertical to 132 is outside the triangle. The angle supplementary to 132 is 48. The angle vertical to THAT is inside the triangle? No.
Let's look at the top intersection.
Line 1: Triangle side.
Line 2: Transversal.
Angle $132$ is between Transversal (top part) and Triangle Side (right part extended?).
Actually, simpler view:
Angle $e$ and $132$ are supplementary linear pair. $e = 48^\circ$.
Angle $d$ and $e$ are vertical angles? No.
Angle $d$ is inside the triangle.
Angle $e$ is "above" the triangle.
If the line forming the left side of the triangle continues straight up, then $d$ and $e$ are supplementary?
Let's stick to the Triangle Sum method, it's robust.
Triangle Angles:
1. Left: $90^\circ$.
2. Bottom: $180 - 138 = 42^\circ$.
3. Top: $180 - 90 - 42 = 48^\circ$.

So, the interior angle at the top is $48^\circ$.
In the diagram, $d$ is the interior angle at the top. So $d = 48^\circ$.
$e$ is adjacent to $d$ on the straight line forming the triangle side?
The line with the arrow goes through the vertex.
Angle $132$ and Angle $e$ are on a straight line?
$e$ is labeled in the space adjacent to $d$.
If $d=48$, and the line is straight, the angle adjacent to $d$ is $132$.
The label $132$ is there.
The label $e$ is vertically opposite to the interior angle?
Let's look at $e$'s position. It is in the V shape above the triangle.
It is vertically opposite to the interior angle $d$.
So $e = 48^\circ$.

Now for $a, b, c$ at the left vertex.
The lines are perpendicular.
Angle $b$: Inside the triangle? No, $b$ is in the top-left quadrant formed by the perpendicular lines.
The triangle is in the bottom-right quadrant relative to that intersection?
No, the triangle is to the right and down.
So the interior angle ($90^\circ$) is in the bottom-right quadrant of the intersection.
Angle $a$: In the bottom-left quadrant?
Angle $b$: In the top-left quadrant?
Angle $c$: In the top-right quadrant?

Let's look at the arcs.
Arc $a$: Bottom-Left.
Arc $b$: Top-Left.
Arc $c$: Top-Right? No, $c$ is at the bottom vertex.

Let's re-read the labels for 5.
Left Vertex: Labels $a$ and $b$.
Bottom Vertex: Label $c$.
Top Vertex: Labels $d$ and $e$.

At Left Vertex:
Perpendicular lines.
Triangle is in one quadrant (say, Q4). Interior angle = 90.
$b$ is in Q2 (Top-Left). Vertically opposite to the interior angle?
If so, $b = 90^\circ$.
$a$ is in Q3 (Bottom-Left). Adjacent to interior angle on straight line?
So $a = 90^\circ$.

At Bottom Vertex:
Exterior angle $138^\circ$.
Interior angle is $42^\circ$.
Label $c$ is the interior angle?
The arc for $c$ is inside the triangle.
So $c = 42^\circ$.

Let's double check $a$ and $b$.
Usually, $a$ and $b$ are not just 90.
Is there a line passing through?
Yes, the hypotenuse passes through the left vertex? No, the legs do.
The left vertex is the intersection of the two legs.
Are the legs straight lines extending through the vertex?
The diagram shows arrows on the ends of the lines forming the "X" at the left.
So yes, they are straight lines.
Therefore, the vertical angle to the interior $90^\circ$ is $b$. So $b=90^\circ$.
The supplementary angle to the interior $90^\circ$ is $a$. So $a=90^\circ$.

Answers for 5:
$a = 90^\circ$
$b = 90^\circ$
$c = 42^\circ$
$d = 48^\circ$
$e = 48^\circ$ (vertically opposite to d)

6.
Triangle with exterior angles.
* Angle $a$: Exterior angle is $110^\circ$. Interior angle $a$ is supplementary.
$$a = 180^\circ - 110^\circ = \mathbf{70^\circ}$$
* Angle $b$: Exterior angle is $100^\circ$. Interior angle $b$ is supplementary.
$$b = 180^\circ - 100^\circ = \mathbf{80^\circ}$$
* Angle $c$: This is the third interior angle.
Sum of interior angles = $180^\circ$.
$$70^\circ + 80^\circ + c = 180^\circ$$
$$150^\circ + c = 180^\circ$$
$$c = \mathbf{30^\circ}$$
*(Check: The exterior angle at the top would be $180-30=150$. The sum of exterior angles is $110+100+150=360$. Correct.)*

---

Part 2: Fill in each blank with a true statement

7.
* Supplement of $\angle B$ is $22^\circ$.
* This means $m\angle B + 22^\circ = 180^\circ$.
* $m\angle B = 180^\circ - 22^\circ = 158^\circ$.
* Since $\angle A \cong \angle B$, $m\angle A = m\angle B$.
* $m\angle A = \mathbf{158^\circ}$.

8.
* $\angle P$ is a right angle ($90^\circ$).
* $\angle P$ and $\angle Q$ are supplementary (add to $180^\circ$).
* $90^\circ + m\angle Q = 180^\circ$.
* $m\angle Q = \mathbf{90^\circ}$.

9.
* $\angle S$ and $\angle T$ are complementary ($S+T=90$).
* $\angle T$ and $\angle U$ are supplementary ($T+U=180$).
* From the first equation, $T < 90$ (assuming positive angles).
* From the second equation, $U = 180 - T$.
* Since $T < 90$, $180 - T > 90$.
* An angle greater than $90^\circ$ (and less than 180) is obtuse.
* Answer: obtuse

10.
* An obtuse angle is between $90^\circ$ and $180^\circ$.
* Let the angle be $x$. $90 < x < 180$.
* Its supplement is $180 - x$.
* If $x > 90$, then $180 - x < 90$.
* An angle less than $90^\circ$ is acute.
* Answer: acute

──────────────────────────────────────

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

7. 158°
8. 90°
9. obtuse
10. acute