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McGraw Hill Integrated II, 2012
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McGraw Hill Integrated II, 2012 View details
4. Proving Triangles Congruent-ASA, AAS
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Exercise 9 Page 368

Use the definition of the midpoint of a segment and the Alternate Interior Angles Theorem. Notice that there is a pair of vertical angles.

Statements
Reasons
1.
UY∥ XW
1.
Given
2.
∠ VYU ≅ ∠ VWX
2.
Alternate Interior Angles Theorem
3.
V is the midpoint of YW
3.
Given
4.
VY ≅ VW
4.
Definition of midpoint
5.
∠ UVY ≅ ∠ XVW
5.
Vertical Angles Theorem
6.
△ UVY ≅ △ XVW
6.
Angle-Side-Angle (ASA) Congruence Postulate
Practice makes perfect

Since V is the midpoint of YW, we get VY ≅ VW. Additionally, UY∥ XW and YW is a transversal, and so by the Alternate Interior Angles Theorem we get ∠ VYU ≅ ∠ VWX.

Notice that ∠ UVY and ∠ XVW are vertical angles which, by the Vertical Angles Theorem, we have that ∠ UVY ≅ ∠ XVW.

Therefore, by the Angle-Side-Angle (ASA) Congruence Postulate we conclude that △ UVY ≅ △ XVW. We summarize the proof in the following two-column table.

Statements
Reasons
1.
UY∥ XW
1.
Given
2.
∠ VYU ≅ ∠ VWX
2.
Alternate Interior Angles Theorem
3.
V is the midpoint of YW
3.
Given
4.
VY ≅ VW
4.
Definition of midpoint
5.
∠ UVY ≅ ∠ XVW
5.
Vertical Angles Theorem
6.
△ UVY ≅ △ XVW
6.
Angle-Side-Angle (ASA) Congruence Postulate
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arrow_right Exercises p.367-371
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Exercises 2 (Page 367)
Exercises 3 (Page 367)
Exercises 4 (Page 367)
Exercises 5 (Page 368)
Exercises 6 (Page 368)
Exercises 7 (Page 368)
Exercises 8 (Page 368)
Exercises 9 (Page 368)
Exercises 10 (Page 368)
Exercises 11 (Page 368)
Exercises 12 (Page 369)
Exercises 13 (Page 369)
Exercises 14 (Page 369)
Exercises 15 (Page 369)
Exercises 16 (Page 369)
Exercises 17 (Page 370)
Exercises 18 (Page 370)
Exercises 19 (Page 370)
Exercises 20 (Page 370)
Exercises 21 (Page 370)
Exercises 22 (Page 370)
Exercises 23 (Page 370)
Exercises 24 (Page 370)
Exercises 25 (Page 370)
Exercises 26 (Page 370)
Exercises 27 (Page 371)
Exercises 28 (Page 371)
Exercises 29 (Page 371)
Exercises 30 (Page 371)
Exercises 31 (Page 371)
Exercises 32 (Page 371)
Exercises 33 (Page 371)
34 engineering
35 engineering
36 engineering
Exercises 37 (Page 371)
Exercises 38 (Page 371)