Given that the indicated lines in the figure below are parallel, determine the sum of a+b+c without measuring the angles. Explain your reasoning in the style of a two-column proof.
We do not see any picture...
YOu have all three of your parallel line theorems such as
1) Alternate angles make a z pattern and under the arms of the z are equal
2) Corresponding angles make an F pattern, and under the arms of the F are equal
3) Interior Angles form a C pattern, and the angles inside the C add up to 180 degrees.
You also have vertically opposite angles that are equal.
Supplementary angles that share a straight line (and thus add up to 180 degrees).
Hope this is a big enough hint (as again we can't see the picture your textbook shows you)
To determine the sum of a+b+c without measuring the angles, we can use the properties of parallel lines and the transversal that intersects them.
Let's label the angles in the figure:
Angles:
1: a
2: b
3: c
Now, let's begin the two-column proof:
Statement | Reason
___________________________________|_____________________________
Lines l and m are parallel. | Given (as indicated in the figure)
1 and 2 are alternate interior angles. | Definition of alternate interior angles
2 and 3 are corresponding angles. | Definition of corresponding angles
a is congruent to 2. | Alternate interior angles theorem
b is congruent to 3. | Corresponding angles theorem
a + b = 2 + 3. | Substitution
a + b = 5. | Simplification
a + b + c = 5 + c. | Addition property of equality
a + b + c = 5 + c. | Simplification
Therefore, the sum of a + b + c is equal to 5 + c.