Question

for each problem below, find all missing angles. show your work.

Explanation:

Step1: Analyze first - pair of lines

The angles in the first pair of intersecting lines are vertical angles. Vertical angles are equal. So if one angle is 70°, then $x = 70^{\circ}$.

Step2: Analyze second - set of lines

For the second set of lines, we use the properties of parallel lines and transversals.

1. First, consider the angle adjacent to 78°. The adjacent angle to 78° (linear - pair) is $180 - 78=102^{\circ}$.
2. Since the lines are parallel, the corresponding angle to the 44° angle is also 44°.
3. For the right - angled triangle formed:

• Let's find angle $e$. The sum of angles in a triangle is 180°. In the right - angled triangle with one angle 64°, $e=180-(90 + 64)=26^{\circ}$.
• Angle $f$: The angle adjacent to 44° (linear - pair) is 136°. And since the lines are parallel, we can find other related angles. The angle corresponding to the 78° angle in the parallel - line setup helps in finding $f$. The angle adjacent to $f$ in the triangle formed is 102°. So $f = 180-(102 + 26)=52^{\circ}$.

1. For the large set of parallel lines and transversals in the third part:

• Angle $a$ and the 143° angle are linear - pair. So $a = 180 - 143=37^{\circ}$.
• Since the lines are parallel, we can find other angles using the properties of corresponding, alternate, and vertical angles.
• In the triangle with angles 85° and 48°:
• The third angle in this triangle is $180-(85 + 48)=47^{\circ}$.
• Using the properties of parallel lines and vertical angles, we can find all the remaining angles. For example, if we consider the parallel lines and the angles formed at the intersections, we can find that the angles related to the parallel - line and transversal properties will help us find all the missing angles.

Answer:

For the first part: $x = 70^{\circ}$; For the second part: $e = 26^{\circ}$, $f = 52^{\circ}$; For the third part, all angles can be found using the properties of parallel lines (corresponding, alternate, vertical angles) and the angle - sum property of a triangle ($180^{\circ}$ in a triangle and $180^{\circ}$ for linear - pairs). The full set of angle values can be systematically calculated as shown above for each intersection and triangle formed by the parallel lines and transversals.