Let's address both diagrams one by one:

First Diagram (Three lines crossing at a point):

• We are given the angles involving $a$, specifically:

  • One angle is $a$,
  • Another is $a + 20^\circ$,
  • And the third angle is $2a$.

• Since all three lines cross at a single point, the angles around that point must add up to $360^\circ$.

Thus, the equation is: $a + (a + 20^\circ) + 2a = 360^\circ$ Simplifying the equation: $a + a + 20^\circ + 2a = 360^\circ$ $4a + 20^\circ = 360^\circ$ $4a = 360^\circ - 20^\circ$ $4a = 340^\circ$ $a = \frac{340^\circ}{4} = 85^\circ$

So, the value of $a$ is $85^\circ$.

Second Diagram (Angle $m$ in a quadrilateral):

• We are given the angles $142^\circ$, $156^\circ$, and $m$. In any quadrilateral, the sum of the interior angles is $360^\circ$.

Thus, the equation is: $m + 142^\circ + 156^\circ = 360^\circ$ Simplifying: $m + 298^\circ = 360^\circ$ $m = 360^\circ - 298^\circ = 62^\circ$

So, the value of $m$ is $62^\circ$.

Would you like more details or have any questions about these calculations? Here are some related questions:

1. What is the general rule for the sum of angles when multiple lines intersect?
2. How do you solve equations involving angles in a polygon?
3. How would you approach similar problems in different geometric contexts?
4. Can you apply this reasoning to solve for angles in other geometric shapes?
5. What are other relationships between angles formed by intersecting lines?

Tip: Always remember that the sum of angles around a point is $360^\circ$, and the sum of interior angles in any polygon is calculated using $(n - 2) \times 180^\circ$, where $n$ is the number of sides.