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McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
5. Parts of Similar Triangles
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Exercise 25 Page 506

Use the Triangle Proportionality Theorem. Then, show that ∠ E≅ ∠ 3 and use the Converse of Isosceles Triangle Theorem.

Statements
Reasons
1.
CD bisects ∠ ACB
By construction, AE∥CD
1.
Given
2.
AD/DB = EC/BC
2.
Triangle Proportionality Theorem
3.
∠ 1 ≅ ∠ 2
3.
Definition of angle bisector
4.
∠ 3 ≅ ∠ 1
4.
Alternate Interior Angles Theorem
5.
∠ 2 ≅ ∠ E
5.
Corresponding Angles Theorem
6.
∠ 3 ≅ ∠ E
6.
Transitive Property of Congruence
7.
EC≅AC
7.
Converse of Isosceles Triangle Theorem
8.
EC=AC
8.
Definition of congruent angles
9.
AD/DB = AC/BC
9.
Substitution
Practice makes perfect

We are asked to write a two-column proof of the given theorem.

Theorem 7.11 Triangle Angle Bisector

An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

First, let's consider △ ABC and let CD be the angle bisector of ∠ C. By definition of a bisector, we have ∠ 1≅ ∠ 2.

Our goal is to prove that ADDB= ACBC. To do so, we are going to draw a line parallel to CD passing through vertex A.

Next, we will extend BC such that it intersects the line drawn before.

By the Triangle Proportionality Theorem we get the following proportion. AD/DB = EC/BC Notice that, by the Corresponding Angles Theorem, ∠ 2 and ∠ E are congruent. Also, we know that CD ∥ EA. Using the Alternate Interior Angles Theorem, we have that ∠ 1≅ ∠ 3. Now, by the Transitive Property of Congruence, ∠ 3 and ∠ E are congruent.

The Converse of Isosceles Triangle Theorem tells us that if two angles in a triangle are congruent, then the sides opposite them are congruent. As a result, we get that EC≅ AC — that is, EC=AC. Finally, we can substitute this fact into the proportion written earlier. AD/DB = EC/BC ⇒ AD/DB = AC/BC

Two-Column Proof

Let's summarize the proof we wrote in the following two-column table. Given: & CD bisects∠ ACB & By construction, AE∥CD Prove: & ADDB = ACBC

Statements
Reasons
1.
CD bisects ∠ ACB
By construction, AE∥CD
1.
Given
2.
AD/DB = EC/BC
2.
Triangle Proportionality Theorem
3.
∠ 1 ≅ ∠ 2
3.
Definition of angle bisector
4.
∠ 3 ≅ ∠ 1
4.
Alternate Interior Angles Theorem
5.
∠ 2 ≅ ∠ E
5.
Corresponding Angles Theorem
6.
∠ 3 ≅ ∠ E
6.
Transitive Property of Congruence
7.
EC≅AC
7.
Converse of Isosceles Triangle Theorem
8.
EC=AC
8.
Definition of congruent angles
9.
AD/DB = AC/BC
9.
Substitution
Subchapter links expand_more
arrow_right Check Your Understanding p.504-505
1
2
3
4
5
arrow_right Practice and Problem Solving p.505-507
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Practice and Problem Solving 9 (Page 505)
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11
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Practice and Problem Solving 19 (Page 506)
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Practice and Problem Solving 25 (Page 506)
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arrow_right H.O.T. Problems p.507
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H.O.T. Problems 33 (Page 507)
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arrow_right Standardized Test Practice p.508
Standardized Test Practice 36 (Page 508)
Standardized Test Practice 37 (Page 508)
38
39
arrow_right Spiral Review p.508
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Spiral Review 43 (Page 508)
Spiral Review 44 (Page 508)
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arrow_right Skills Review p.508
Skills Review 46 (Page 508)
Skills Review 47 (Page 508)
Skills Review 48 (Page 508)
Skills Review 49 (Page 508)
Skills Review 50 (Page 508)
Skills Review 51 (Page 508)