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McGraw Hill Integrated II, 2012 View details

3. Similar Triangles

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Exercise 27 Page 567

Use the Side-Angle-Side (SAS) Similarity Theorem.

Statements	Reasons
1. △ XYZ and △ ABC are right triangles	1. Given
2. ∠ Y and ∠ B are right angles	2. Definition of right triangle
3. ∠ Y ≅ ∠ B	3. All right angles are congruent
4. XY/AB=YZ/BC	4. Given
5. △ YXZ ~ △ BAC	5. SAS Similarity Theorem

Practice makes perfect

We are given two right triangles that have two pairs of proportional corresponding sides. Also, by the definition of right triangles, ∠ Y and ∠ B are right angles.

Notice that the sides that are proportional are the legs of the triangles, and since all right angles are congruent, we have that ∠ Y ≅ ∠ B.

∠ Y ≅ ∠ B and XY/AB = YZ/BC Therefore, by the Side-Angle-Side (SAS) Similarity Theorem △ YXZ ~ △ BAC, which is what we wanted to prove.

Two-Column Proof

Given: & △ XYZ and △ ABC are right & triangles; XYAB= YZBC Prove: & △ YXZ ~ △ BAC Let's summarize the proof we did above in the following two-column table.

Statements	Reasons
1. △ XYZ and △ ABC are right triangles	1. Given
2. ∠ Y and ∠ B are right angles	2. Definition of right triangle
3. ∠ Y ≅ ∠ B	3. All right angles are congruent
4. XY/AB=YZ/BC	4. Given
5. △ YXZ ~ △ BAC	5. SAS Similarity Theorem

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Exercises p.565-569

Exercise 27 Page 567

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Two-Column Proof