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Problem 16:

Statement: The difference in measures of two complementary angles is $16^\circ$. Find the measures of the angles.

Solution: Let the two complementary angles be $x$ and $y$. Since they are complementary: $x + y = 90^\circ$ The difference between the angles is given as $16^\circ$: $x - y = 16^\circ$ Now, solve these two equations simultaneously.

1. Add the equations: $(x + y) + (x - y) = 90^\circ + 16^\circ$ $2x = 106^\circ \quad \Rightarrow \quad x = 53^\circ$

2. Substitute $x = 53^\circ$ into the first equation: $53^\circ + y = 90^\circ \quad \Rightarrow \quad y = 37^\circ$

So, the two angles are $53^\circ$ and $37^\circ$.

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Problem 17:

Statement: In the given figure, if the value of $\angle L = 40^\circ$, find the measures of all the remaining angles. Mention the properties used.

Solution: Given $\angle L = 40^\circ$.

Since $\overline{PQ}$ and $\overline{RS}$ are parallel lines, and $\overline{LT}$ is the transversal, the following properties can be used:

• Corresponding Angles: These are equal when a transversal intersects two parallel lines.
• Alternate Interior Angles: These are equal when a transversal intersects two parallel lines.
• Vertically Opposite Angles: These are equal.

Using the properties:

1. $\angle Q = \angle L = 40^\circ$ (Corresponding angles)
2. $\angle M = \angle L = 40^\circ$ (Vertically opposite angles)
3. $\angle P = \angle R = \angle S = \angle T = 140^\circ$ (Using the linear pair property, $180^\circ - 40^\circ$)

Thus, the angles are: $\angle Q = 40^\circ, \quad \angle M = 40^\circ, \quad \angle P = 140^\circ, \quad \angle R = 140^\circ, \quad \angle S = 140^\circ, \quad \angle T = 140^\circ$

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Problem 18:

Statement: Find the values of $x$, $y$, and $z$ in the given figure.

Solution: From the figure:

• $\angle x + 55^\circ = 180^\circ$ (Linear pair)

• $x = 180^\circ - 55^\circ = 125^\circ$

• $\angle y = \angle z = 55^\circ$ (Vertically opposite angles)

Thus: $x = 125^\circ, \quad y = 55^\circ, \quad z = 55^\circ$

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Would you like any further details or explanations? Here are some related questions:

1. What are supplementary angles, and how do they differ from complementary angles?
2. How do we determine if two lines are parallel using angle properties?
3. Can you explain the concept of vertically opposite angles in detail?
4. What is the significance of the transversal line in angle relationships?
5. How do alternate interior angles help in proving lines parallel?
6. Could you explain the relationship between linear pairs and straight angles?
7. What are corresponding angles, and where do they appear in geometry?
8. How can angle relationships be used to find unknown angles in complex figures?

Tip: Always remember that when two lines are intersected by a transversal, various angle relationships like corresponding, alternate interior, and alternate exterior angles can help determine unknown angles.