Chapter 4: Q8. (page 133)

Write proofs in two-column form.
Given: $\bar{A D} ∥ \bar{C B}; \bar{A D} ≅ \bar{C B}$
Prove: $\bar{D C} ∥ \bar{A B}$

Short Answer

Expert verified

Statement	Reason
1. $\bar{C D} ≅ \bar{A B}$	Given
2. $∠ 3 ≅ ∠ 4$	Alternate interior angles
3. $\bar{B D} ≅ \bar{B D}$	Common line segment
4. $Δ C B D ≅ Δ A D B$	SAS congruency criteria
5. $∠ 1 ≅ ∠ 2$	Corresponding parts of congruent triangle
6. $\bar{D C} ∥ \bar{A B}$	If alternate interior angles are equal then lines are parallel

Step by step solution

Step 1. Show that BD¯≅BD¯.

Line segment $\bar{B D}$ is common in both triangles, by reflexive property of congruency $\bar{B D}$ is congruent to itself

$\bar{B D} ≅ \bar{B D}$

Step 2. Show that ∠3≅∠4.

Since $∠ 3$ and $∠ 4$ forms a pair of alternate interior angles

So, $∠ 3 ≅ ∠ 4$

Step 3. Show that ΔCBD≅ΔADB.

From step 1 and 2 and given

$Δ C B D ≅ Δ A D B$ by SAS (Side-Angle-Side) congruency criteria

Step 4. Show that ∠1≅∠2.

Since, corresponding parts of congruent triangle are congruent

So, $∠ 1 ≅ ∠ 2$

Step 5. Show that DC¯∥AB¯.

Since, $∠ 1 and ∠ 2$ forms a pair of congruent alternate interior angles. So, lines are parallel.

$\bar{D C} ∥ \bar{A B}$

Thus, two column proof is

Statement	Reason
1. $\bar{C D} ≅ \bar{A B}$	Given
2. $∠ 3 ≅ ∠ 4$	Alternate interior angles
3. $\bar{B D} ≅ \bar{B D}$	Common line segment
4. $Δ C B D ≅ Δ A D B$	SAS congruency criteria
5. $∠ 1 ≅ ∠ 2$	Corresponding parts of congruent triangle
6. $\bar{D C} ∥ \bar{A B}$	If alternate interior angles are equal then lines are parallel

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Write proofs in two-column form.
Given: $\bar{A D} ∥ \bar{C B}; \bar{A D} ≅ \bar{C B}$
Prove: $\bar{D C} ∥ \bar{A B}$

Short Answer

Step by step solution

Step 1. Show that BD¯≅BD¯.

Step 2. Show that ∠3≅∠4.

Step 3. Show that ΔCBD≅ΔADB.

Step 4. Show that ∠1≅∠2.

Step 5. Show that DC¯∥AB¯.

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Write proofs in two-column form.Given:AD¯∥CB¯;AD¯≅CB¯Prove:DC¯∥AB¯

Short Answer

Step by step solution

Step 1. Show that BD¯≅BD¯.

Step 2. Show that ∠3≅∠4.

Step 3. Show that ΔCBD≅ΔADB.

Step 4. Show that ∠1≅∠2.

Step 5. Show that DC¯∥AB¯.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Write proofs in two-column form.
Given: $\bar{A D} ∥ \bar{C B}; \bar{A D} ≅ \bar{C B}$
Prove: $\bar{D C} ∥ \bar{A B}$