Exercise 27 Page 589

Let △ QTS and △ XWZ be a pair of triangles such that they are similar. Also, let TR and WY be angle bisectors.

Because corresponding angles of similar triangles are congruent, we have the following pair of congruences. ∠ T≅ ∠ W and ∠ Q ≅ ∠ X ⇓ m∠ T = m∠ W and m∠ Q = m∠ XBy the definition of an angle bisector we have that m∠ RTQ = 12m∠ T and m∠ YWX = 12m∠ W.

From the above, ∠ RTQ ≅ ∠ YWX.

Then, △ TRQ ~ △ WYX because of the Angle-Angle (AA) Similarity Postulate. This leads us to set the required proportion. TR/WY = QT/XW ✓

Two-Column Proof

Given: & △ QTS ~ △ XWZ, TR and WY & are angle bisectors Prove: & TRWY = QTXW We will summarize the proof we did above in the following two-column table.