Exercise 123 Page 450

We will create either a flowchart or a two-column proof for the conclusion in each of the Parts A, B, and C of the previous exercise.

Part A

From the diagram in Part A from the previous exercise we have two pieces of given information.

In addition to this we can identify a pair of vertical angles at ∠ C. This means we can use the ASA (Angle-Side-Angle) Congruence Theorem to say the two triangles are congruent. Now we have all the information we need to make a two-column proof for Part A.

Part B

The diagram in Part B gives us three pieces of information.

We know that EF is congruent to CB and GF is congruent to DB. In addition, ∠ F and ∠ B are both right angles. Since they have equal measure, they are congruent. Therefore we can use the SAS (Side-Angle-Side) Congruence Theorem to say that the two triangles are congruent. Let's illustrate this using a flowchart.

Part C

This time, the diagram gives us two pieces of information.

We know that KJ and IJ are congruent and that LK and HI are congruent. We can also see that △ LJI and △ HJK share ∠ J. Let's separate the two triangles to see the situation better.

Even when we do not know the exact lengths of segments, we can use the Segment Addition Postulate to show relationships. JH=JI+IH JL=JK+KL Because of the given congruency markings, we know that JI ≅ JK and IH ≅ KL. This means that they also have equal lengths and that JH ≅ JL. We also know that ∠ J is congruent to itself by the Reflexive Property of Congruence.

This tells us that △ LJI and △ HJK have two pairs of congruent sides, and the angles between the sides are congruent as well. Therefore △ LJI and △ HJK are congruent by the SAS (Side-Angle-Side) Congruence Theorem. Let's make a two-column proof!