Q11. 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: Q11. WE (page 137)

Write proofs in two鈥揷olumn form.
Theorem 4鈥�1.

Short Answer

Expert verified

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

Step by step solution

Step 1. Draw an auxiliary line through angle A.

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. SAS postulate.

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

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

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

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\overset{炉}{A B} 鈮� \overset{炉}{A C}$	Given
$鈭� B A D 鈮� 鈭� C A D$	Angle bisector theorem
$\overset{炉}{A D} 鈮� \overset{炉}{A D}$	Reflexive property
$螖 B A D 鈮� 螖 C A D$	SAS postulate
$鈭� B 鈮� 鈭� C$	Congruent angles of congruent triangles

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91影视

Write proofs in two鈥揷olumn form.
Theorem 4鈥�1.

Short Answer

Step by step solution

Step 1. Draw an auxiliary line through angle A.

Step 2. Apply angle bisector theorem.

Step 3. Reflexive property.

Step 4. SAS postulate.

Step 5. Two column proof.

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91影视

Write proofs in two鈥揷olumn form.Theorem 4鈥�1.

Short Answer

Step by step solution

Step 1. Draw an auxiliary line through angle A.

Step 2. Apply angle bisector theorem.

Step 3. Reflexive property.

Step 4. SAS postulate.

Step 5. Two column proof.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Write proofs in two鈥揷olumn form.
Theorem 4鈥�1.