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McGraw Hill Integrated II, 2012
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McGraw Hill Integrated II, 2012 View details
3. Similar Triangles
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Exercise 28 Page 567

Use the Alternate Interior Angles Theorem and the Angle-Angle (AA) Similarity Postulate.

Statements
Reasons
1.
ABCD is a trapezoid
1.
Given
2.
AB∥ CD
2.
Definition of trapezoid
3.
∠ CAB ≅ ∠ ACD and ∠ ABD ≅ ∠ CDB
3.
Alternate Interior Angles Theorem
4.
△ DPC ~ △ BPA
4.
AA Similarity Postulate
5.
DP/PB = CP/PA
5.
Definition of similar triangles
Practice makes perfect

Since ABCD is a trapezoid, we have that AB∥ CD.

By the Alternate Interior Angles Theorem we get that ∠ CAB ≅ ∠ ACD and ∠ ABD ≅ ∠ CDB.

The Angle-Angle (AA) Similarity Postulate allows us to conclude that △ DPC ~ △ BPA. Finally, by the definition of similar triangles we can write the following proportions. DP/PB = CP/PA ✓

Two-Column Proof

Given: & ABCD is a trapezoid Prove: & DPPB = CPPA Let's summarize the proof we did above in the following two-column table.

Statements
Reasons
1.
ABCD is a trapezoid
1.
Given
2.
AB∥ CD
2.
Definition of trapezoid
3.
∠ CAB ≅ ∠ ACD and ∠ ABD ≅ ∠ CDB
3.
Alternate Interior Angles Theorem
4.
△ DPC ~ △ BPA
4.
AA Similarity Postulate
5.
DP/PB = CP/PA
5.
Definition of similar triangles
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Exercises 2 (Page 565)
Exercises 3 (Page 565)
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