CPM Homework Help : CCG Problem 6-46

2 horizontal parallel lines & a triangle positioned, with the bottom side on the bottom parallel, & the top vertex on the top parallel, Angles labeled as follows: below top parallel, left of triangle, d, in triangle, c, right of triangle, e, above bottom parallel, left of triangle, f, left in triangle, a, right in triangle, b, right of triangle, g. Copy the diagram below onto your paper. If $a=53^{\circ}$ and $g=125^{\circ}$ , find the measures of each labeled angle. Explain how you find each angle, citing definitions and theorems from your toolkit that support your steps. Remember that you can find the angles in any order, depending on the angle relationships you use.

Hint:

You can find the angles in any order.

Step 1:

Since $∠a$ and $∠f$ are adjacent supplementary angles (linear pair), then we can solve this algebraically.
$\begin{array}{r} m \angle a + m \angle f = 180° \\ 53° + m \angle f = 180° \\ m \angle f = 127° \end{array}$

Step 2:

$∠b$ and $∠g$ are another linear pair.
They also are supplementary.
$\begin{array}{r} m∠b + m∠g = 180º \\ m∠b + 125º \ = 180º \\ m∠b = 55º \end{array}$

More Help:

If parallel lines are cut by a transversal, then alternate interior angles are congruent.

Step 3:

$\begin{array}{l} m∠a=m∠d=53º \\ m∠b=m∠e=55º \end{array}$

Step 4:

Using the Triangle Angle Sum theorem:
$\begin{array}{r} m∠a + m∠b + m∠c = 180º \\ 53º + 55º + m∠c = 180º \\ m∠c = 72º \end{array}$

Answer:

$a=53º$ , $b=55º$ , $c=72º$ , $d=53º$ , $e=55º$ , $f=127º$ , $g=125º$