Answer:
11. x=8, y=30
12. w=63°, x=86°, y=38°, z=25°
13. x=18, y=15
Explanation:
You have been studying the angles related to transversals and parallel lines, so you know alternate (interior or exterior) angles are congruent (equal) and same-side (interior or exterior) angles are supplementary. You also know that linear angles sum to 180°, the measure of an angle formed by a straight line.
The red arrows in these diagrams signify that the lines so marked are parallel, so the above relationships apply.
11. The diagonal line is a transversal of the top and bottom parallel lines, so the angle marked 5x° is equal to the angle marked 40°. Writing that relationship as an equation, you have ...
... 5x° = 40° . . . . . . . the relationship of the angles
... x = 40/5 = 8 . . . . divide the equation by 5° to find x
The angle marked 3y° and the angle marked with a right-angle symbol are same-side interior angles formed by the vertical transversal with the top and bottom horizontal lines. Hence those angles are supplementary.
... 3y° +90° = 180°
... 3y° = 90° . . . . . . . subtract 90°
... y = 90/3 = 30 . . . .divide by 3°
12. These problems make use of the sum of angles that form a line (180°) and the sum of angles in a triangle (180°).
Together 117° and w form a line, so ...
... 117° + w = 180°
... w = 63° . . . . . . . . . . subtract 117°
The angles of a triangle add to give 180°, so ...
... w + 31° + x = 180°
... 63° +31° + x = 180° . . . . . put in the known value for w
... x = 180° -63° -31°
... x = 86°
Together, x and 56° and y form a line, so the sum of their measures is 180°.
... x + 56° + y = 180°
... 86° + 56° + y = 180° . . . . put in the known value for x
... y = 180° -86° -56° . . . . . . subtract the constants on the left
... y = 38°
The sum of angles y, 117°, and z is 180°.
... 38° + 117° + z = 180° . . . . use the known value for y
... z = 180° -38° -117° . . . . . . subtract the constants on the left
... z = 25°
13. The relationships used in problem 11 also can be used here.
... 5x° + 90° = 180°
... x = (180° -90°)/5°
... x = 18
The angles involving y are same-side interior angles, so are supplementary.
... 5(y +11)° + (4y -10)° = 180°
... 5y +55 + 4y -10 = 180 . . . . . . eliminate parentheses, divide by °
... 9y +45 = 180 . . . . . . . . . . . . . collect terms
... y + 5 = 20 . . . . . . . . . . . . . . . divide by 9
... y = 15 . . . . . . . . . . . . . . . . . . . .subtract 5