Chapter 4: Q11. WE (page 137)

Write proofs in two–column form.
Theorem 4–1.

Short Answer

Expert verified

Statement	Reason
$\bar{AB} ≅ \bar{AC}$	Given
$∠ BAD ≅ ∠ CAD$	Angle bisector theorem
$\bar{AD} ≅ \bar{AD}$	Reflexive property
$Δ BAD ≅ Δ CAD$	SAS postulate
$∠ B ≅ ∠ C$	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 $\bar{A D}$ bisects $∠ A$ .

Therefore, $∠ B A D ≅ ∠ C A D$ .

Step 3. Reflexive property.

Reflexive property states that $\bar{P Q} ≅ \bar{P Q}$ .

Therefore, $\bar{A D} ≅ \bar{A D}$ .

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 $\bar{A B} ≅ \bar{A C}$ , $\bar{A D} ≅ \bar{A D}$ and $∠ B A D ≅ ∠ C A D$ . Therefore, $Δ B A D ≅ Δ C A D$ which implies that $∠ B ≅ ∠ C$ .

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
$\bar{A B} ≅ \bar{A C}$	Given
$∠ B A D ≅ ∠ C A D$	Angle bisector theorem
$\bar{A D} ≅ \bar{A D}$	Reflexive property
$Δ B A D ≅ Δ C A D$	SAS postulate
$∠ B ≅ ∠ C$	Congruent angles of congruent triangles

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Write proofs in two–column form.
Theorem 4–1.

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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Write proofs in two–column form.Theorem 4–1.

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

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Write proofs in two–column form.
Theorem 4–1.