Answer:
7g. y = x + z
7i. x +y +z = 360°
Explanation:
You want equations connecting x, y, z in the given diagrams involving parallel lines.
G
The attachment shows one way of looking at the relationship.
By extending the line from angle x to meet line CD, we form a triangle. The angle we formed at line CD is an alternate interior angle relative to angle x, so has the same measure. In the triangle formed, this angle and angle z are "remote interior angles" with respect to exterior angle y.
The exterior angle theorem tells us ...
y = x + z
I
This geometry is substantially identical to the one in figure G, except that angle y is now on the other side of the lines that form it. Comparing to figure G, we see that the angle we have marked y' corresponds to the angle y in figure G. Angles y and y' in figure I together make a full circle of 360°.
Using the above relation, we have ...
y' = x + z
360° -y = x + z
x + y + z = 360°
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Additional comment
If you extend both of the lines that form angle y so they make an X, and draw a horizontal line through that parallel to the other two lines, then you can perhaps see the relationships more directly.
In Figure G, alternate interior angles corresponding to x and z will add up to form angle y.
In Figure I, angles corresponding to x and z together with angle y will complete the circle around the vertex of y.
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