Let's go through the math problems from the image.

6) Look for Relationships

The problem involves a square with lines $a$, $b$, and $c$, and you're asked to find the measure of $\angle 1$ in order for line $c$ to be parallel to line $b$.

• Given: $\angle a = 74^\circ$
• Unknown: $\angle 1$ for line $c$ to be parallel to line $b$.

Explanation: In order for line $c$ to be parallel to line $b$, $\angle 1$ should correspond to the angle created by alternate interior angles with line $a$ and line $b$. So, the sum of $\angle a$ and $\angle 1$ must equal $180^\circ$ (since they form a linear pair).

$\angle 1 = 180^\circ - 74^\circ = 106^\circ$

So, $\angle 1 = 106^\circ$.

7) Make Sense and Persevere

Malia draws diagonals on a square, and you're asked to find the angle measures to make lines parallel.

• Part a: What should $\angle 1$ be for line $b$ to be parallel to line $a$?

Given that $\angle 1 = 125^\circ$, the other angle on the same line would be $180^\circ - 125^\circ = 55^\circ$. This means $\angle 1$ needs to be $125^\circ$ to keep the diagonal lines parallel.

• Part b: What should $\angle 2$ be for line $c$ to be parallel to line $b$?

For lines $b$ and $c$ to be parallel, $\angle 2$ should equal the corresponding angle created by the intersection of line $a$ and $b$. Thus, $\angle 2 = 55^\circ$.

8) SAT/ACT Problem

This is about determining which of the parallel conditions holds true in the diagram.

• Statement A: If $\angle 1 \cong \angle 2$, then $g \parallel h$.

This is false because $\angle 1$ and $\angle 2$ alone do not guarantee that $g \parallel h$.

• Statement B: If $\angle 1 \cong \angle 3$, then $g \parallel h$.

This is true, because corresponding angles are equal when two lines are parallel.

• Statement C: If $\angle 2 \cong \angle 4$, then $h \parallel k$.

This is also true, as corresponding angles being equal implies parallel lines.

• Statement D: If $\angle 3 \cong \angle 2$, then $j \parallel k$.

This is false, since those angles are not in a configuration that implies parallel lines.

Correct answers are:

• B and C are true.

Would you like more details on any specific problem? Here are 5 follow-up questions you could explore:

1. What are the conditions for two lines to be parallel based on angle relationships?
2. How can we apply the alternate interior angle theorem in different shapes?
3. What other geometric properties can be used to determine if two lines are parallel?
4. How do diagonals in squares and rectangles influence the geometry of the shape?
5. Can we use these principles to solve problems involving polygons other than squares?

Tip: When solving problems involving parallel lines and angles, always look for angle pairs like corresponding, alternate interior, or consecutive angles, as they often hold key clues to the solution.