Angles Worksheet 12 with Clues - Calculate missing angles and solve for x and y in geometric figures.

Problem: Solve the missing angles in the given worksheet and find the values of \( x \) and \( y \).

#### Step-by-Step Solution:

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1. Basic Concepts Used:

1. Straight Line: Angles on a straight line sum to \( 180^\circ \).
2. Vertically Opposite Angles: Angles opposite each other when two lines intersect are equal.
3. Corresponding Angles: When a transversal intersects parallel lines, corresponding angles are equal.
4. Alternate Interior Angles: When a transversal intersects parallel lines, alternate interior angles are equal.
5. Sum of Angles in a Triangle: The sum of the interior angles in a triangle is \( 180^\circ \).
6. Isosceles Triangle: Two angles opposite equal sides are equal.
7. Equilateral Triangle: All three angles are \( 60^\circ \).

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Section 1: Calculate the Missing Angles

#### Problem 1:
- Given: \( 52^\circ \)
- To find: The missing angle.

Solution:
- The two angles are vertically opposite angles.
- Vertically opposite angles are equal.
- Therefore, the missing angle is \( 52^\circ \).

Answer: \( 52^\circ \)

#### Problem 2:
- Given: \( 104^\circ \) and \( 117^\circ \)
- To find: The missing angles \( b^\circ \) and \( c^\circ \).

Solution:
- The angle \( b^\circ \) is a corresponding angle to \( 104^\circ \). Corresponding angles are equal.
\[
b = 104^\circ
\]
- The angle \( c^\circ \) is supplementary to \( 117^\circ \) (angles on a straight line sum to \( 180^\circ \)).
\[
c = 180^\circ - 117^\circ = 63^\circ
\]

Answer: \( b = 104^\circ \), \( c = 63^\circ \)

#### Problem 3:
- Given: \( 67^\circ \), \( 75^\circ \), and \( 90^\circ \)
- To find: The missing angles \( d^\circ \) and \( e^\circ \).

Solution:
- The angle \( d^\circ \) is an alternate interior angle to \( 67^\circ \). Alternate interior angles are equal.
\[
d = 67^\circ
\]
- The angle \( e^\circ \) is supplementary to \( 75^\circ \) (angles on a straight line sum to \( 180^\circ \)).
\[
e = 180^\circ - 75^\circ = 105^\circ
\]

Answer: \( d = 67^\circ \), \( e = 105^\circ \)

#### Problem 4:
- Given: An equilateral triangle.
- To find: The missing angle \( f^\circ \).

Solution:
- In an equilateral triangle, all angles are \( 60^\circ \).
\[
f = 60^\circ
\]

Answer: \( f = 60^\circ \)

#### Problem 5:
- Given: An isosceles triangle with one angle \( 124^\circ \).
- To find: The missing angle \( g^\circ \).

Solution:
- In an isosceles triangle, the base angles are equal. Let the base angles be \( g^\circ \).
- The sum of the angles in a triangle is \( 180^\circ \).
\[
g + g + 124^\circ = 180^\circ
\]
\[
2g + 124^\circ = 180^\circ
\]
\[
2g = 56^\circ
\]
\[
g = 28^\circ
\]

Answer: \( g = 28^\circ \)

#### Problem 6:
- Given: A triangle with angles \( 41^\circ \), \( 110^\circ \), and \( h^\circ \).
- To find: The missing angle \( h^\circ \).

Solution:
- The sum of the angles in a triangle is \( 180^\circ \).
\[
h + 41^\circ + 110^\circ = 180^\circ
\]
\[
h + 151^\circ = 180^\circ
\]
\[
h = 29^\circ
\]

Answer: \( h = 29^\circ \)

#### Problem 7:
- Given: A triangle with one angle \( 133^\circ \) and another angle \( 112^\circ \).
- To find: The missing angle \( i^\circ \).

Solution:
- The sum of the angles in a triangle is \( 180^\circ \).
\[
i + 133^\circ + 112^\circ = 180^\circ
\]
\[
i + 245^\circ = 180^\circ
\]
\[
i = -65^\circ
\]
- This result is not possible because angles cannot be negative. There might be a mistake in the problem setup or interpretation. Assuming the problem involves exterior angles or additional context, we need clarification.

#### Problem 8:
- Given: Angles \( 27^\circ \), \( 86^\circ \), and \( j^\circ \).
- To find: The missing angle \( j^\circ \).

Solution:
- The angle \( j^\circ \) is supplementary to \( 86^\circ \) (angles on a straight line sum to \( 180^\circ \)).
\[
j = 180^\circ - 86^\circ = 94^\circ
\]

Answer: \( j = 94^\circ \)

#### Problem 9:
- Given: Angles \( 3^\circ \), \( 20^\circ \), and \( 34^\circ \).
- To find: The missing angle \( k^\circ \).

Solution:
- The angle \( k^\circ \) is supplementary to the sum of the given angles (angles on a straight line sum to \( 180^\circ \)).
\[
k = 180^\circ - (3^\circ + 20^\circ + 34^\circ)
\]
\[
k = 180^\circ - 57^\circ = 123^\circ
\]

Answer: \( k = 123^\circ \)

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Section 2: Extension - Find the Value of \( x \) and \( y \)

#### Problem 10:
- Given: Angles \( x + 10^\circ \) and \( 2x + 20^\circ \).
- To find: The value of \( x \).

Solution:
- These angles are supplementary (angles on a straight line sum to \( 180^\circ \)).
\[
(x + 10^\circ) + (2x + 20^\circ) = 180^\circ
\]
\[
3x + 30^\circ = 180^\circ
\]
\[
3x = 150^\circ
\]
\[
x = 50^\circ
\]

Answer: \( x = 50^\circ \)

#### Problem 11:
- Given: Angles \( 2x + 85^\circ \) and \( 12x - 5^\circ \).
- To find: The value of \( x \).

Solution:
- These angles are vertically opposite angles, so they are equal.
\[
2x + 85^\circ = 12x - 5^\circ
\]
\[
85^\circ + 5^\circ = 12x - 2x
\]
\[
90^\circ = 10x
\]
\[
x = 9^\circ
\]

Answer: \( x = 9^\circ \)

#### Problem 12:
- Given: Angles \( 2x + 10^\circ \) and \( y^\circ \).
- To find: The value of \( y \).

Solution:
- These angles are corresponding angles, so they are equal.
\[
y = 2x + 10^\circ
\]
- From Problem 11, we found \( x = 9^\circ \).
\[
y = 2(9^\circ) + 10^\circ = 18^\circ + 10^\circ = 28^\circ
\]

Answer: \( y = 28^\circ \)

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Final Answers:

\[
\boxed{
\begin{aligned}
&\text{Problem 1: } 52^\circ \\
&\text{Problem 2: } b = 104^\circ, c = 63^\circ \\
&\text{Problem 3: } d = 67^\circ, e = 105^\circ \\
&\text{Problem 4: } f = 60^\circ \\
&\text{Problem 5: } g = 28^\circ \\
&\text{Problem 6: } h = 29^\circ \\
&\text{Problem 7: } \text{(Clarification needed)} \\
&\text{Problem 8: } j = 94^\circ \\
&\text{Problem 9: } k = 123^\circ \\
&\text{Extension: } x = 50^\circ, y = 28^\circ
\end{aligned}
}
\]