Q12. WE Write proofs in two鈥揷olumn for... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 4: Q12. WE (page 137)

Write proofs in two鈥揷olumn form.
Theorem 4鈥�2.

Short Answer

Expert verified

Statement	Reason
$鈭� B 鈮� 鈭� C$	Given
$鈭� BAD 鈮� 鈭� CAD$	Angle bisector theorem
$\overset{炉}{AD} 鈮� \overset{炉}{AD}$	Reflexive property
$螖 BAD 鈮� 螖 CAD$	ASA postulate
$\overset{炉}{AB} 鈮� \overset{炉}{AC}$	Congruent sides of congruent triangles

Step by step solution

Step 1. Draw an angle bisector through angle A.

Figure shows, $螖 A B C$ with an angle bisector at Athat meets $B C$ at pointD.

Step 2. Apply angle bisector theorem.

Angle bisector divides an angle into two equal parts.

From the figure in step 1 it can be observed that $\overset{炉}{A D}$ bisects $鈭� A$ .

Therefore, $鈭� B A D 鈮� 鈭� C A D$ .

Step 3. Reflexive property.

Reflexive property states that $\overset{炉}{P Q} 鈮� \overset{炉}{P Q}$ .

Therefore, $\overset{炉}{A D} 鈮� \overset{炉}{A D}$ .

Step 4. ASA postulate.

If two angles and included side of one triangle are congruent to two angles and included side of other triangle then the triangles are said to be congruent.

Here $鈭� B 鈮� 鈭� C$ , $\overset{炉}{A D} 鈮� \overset{炉}{A D}$ and $鈭� B A D 鈮� 鈭� C A D$ . Therefore, $螖 B A D 鈮� 螖 C A D$ which implies that $\overset{炉}{A B} 鈮� \overset{炉}{A C}$ .

It is proved that if the two angles of a triangle are congruent then the sides opposite to those angles are congruent.

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$鈭� B 鈮� 鈭� C$	Given
$鈭� B A D 鈮� 鈭� C A D$	Angle bisector theorem
$\overset{炉}{A D} 鈮� \overset{炉}{A D}$	Reflexive property
width="111" height="20" role="math">螖BAD鈮�螖CAD	ASA postulate
$\overset{炉}{A B} 鈮� \overset{炉}{A C}$	Congruent sides of congruent triangles

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91影视

Write proofs in two鈥揷olumn form.
Theorem 4鈥�2.

Short Answer

Step by step solution

Step 1. Draw an angle bisector through angle A.

Step 2. Apply angle bisector theorem.

Step 3. Reflexive property.

Step 4. ASA postulate.

Step 5. Two column proof.

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91影视

Write proofs in two鈥揷olumn form.Theorem 4鈥�2.

Short Answer

Step by step solution

Step 1. Draw an angle bisector through angle A.

Step 2. Apply angle bisector theorem.

Step 3. Reflexive property.

Step 4. ASA postulate.

Step 5. Two column proof.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Write proofs in two鈥揷olumn form.
Theorem 4鈥�2.