I'll go through the questions one by one and provide clear solutions.

Q21: The pentagon in the diagram below is formed by line $PQ$. What is the degree measure of angle $y$?

Solution:

The pentagon is irregular, and we can see the following:

• Angle $PQR = 90^\circ$
• Angle $QRS = 90^\circ$
• Angle $RST$ and angle $STP$ are given, but we need to know that the sum of the interior angles of a pentagon is given by the formula:

$\text{Sum of interior angles} = (5-2) \times 180^\circ = 540^\circ$

Let's denote the interior angles of the pentagon as $A, B, C, D, E$.

So,

$A + B + C + D + E = 540^\circ$

Given that $A = 90^\circ$, $B = 90^\circ$, and let's assume $y = D$. We need to solve for $y$ by subtracting the sum of the known angles from 540°.

Without specific values for all angles, we typically find $y$ if other angles are known. However, if all other angles are not given, we'd need that information to determine $y$ accurately. In standard problems like this, the problem should provide or be solvable by complementary angles or symmetry assumptions, but the image does not make that clear.

Therefore, additional data or clarification is needed to solve this directly.

Q22: In the diagram below, two parallel lines are intersected by a transversal. A measure of angle $x = 120^\circ$ is shown. Find the measure of angle $y$.

Solution:

When two parallel lines are intersected by a transversal, the following angles are equal:

• Alternate interior angles
• Corresponding angles

Given $x = 120^\circ$, if $y$ is the corresponding or alternate interior angle, then:

$y = x = 120^\circ$

If $y$ is adjacent to $x$, then it would be:

$y = 180^\circ - x = 180^\circ - 120^\circ = 60^\circ$

The specific relation between $x$ and $y$ depends on the diagram. Based on the parallel line and transversal properties, if $y$ is directly corresponding to $x$, then $y = 120^\circ$.

Q23: Given that the size of the angles of a polygon are $120^\circ, 110^\circ, 135^\circ, x$, and $130^\circ$. Find the value of $x$.

Solution:

The sum of the interior angles of a polygon is given by:

$\text{Sum of interior angles} = (n-2) \times 180^\circ$

where $n$ is the number of sides of the polygon. For this question, $n = 5$ (as there are 5 angles), so:

$\text{Sum of angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$

Now, sum the given angles and set it equal to 540°:

$120^\circ + 110^\circ + 135^\circ + 130^\circ + x = 540^\circ$

Calculate:

$495^\circ + x = 540^\circ$

So,

$x = 540^\circ - 495^\circ = 45^\circ$

Thus, $x = 45^\circ$.

Would you like to see more details, or do you have any questions?

Related Questions:

1. How do you calculate the sum of interior angles for any polygon?
2. What are the properties of angles when a transversal intersects parallel lines?
3. If the sum of angles in a polygon doesn't add up to the theoretical value, what could be the cause?
4. How would the problem change if the polygon were not regular?
5. What are alternate interior angles, and why are they equal?

Tip:

When solving geometry problems involving angles, always identify the type of polygon or lines (parallel, transversal) you're dealing with, as specific rules will simplify your calculations.