Chapter 4: Q14. (page 145)

Write the proofs in two-column form.
Given: $∠ R ≅ ∠ T$ ; $\bar{RS} ∥ \bar{QT}$
Prove: $\bar{RS} ≅ \bar{TQ}$
(Hint: What auxiliary line can you draw to form congruent triangles?)

Short Answer

Expert verified

The proofs in two-column form is:

Statements	Reasons
Join $\bar{Q S}$	By construction
$∠ R ≅ ∠ T$	Given
$\bar{R S} ∥ \bar{T Q}$	Given
$∠ R S Q ≅ ∠ T Q S$	If a line intersects two parallel lines then each pair of alternate angles are equal.
$\bar{Q S} ≅ \bar{Q S}$	Reflexive property
$△ R S Q ≅ △ T Q S$	By AAS congruence
$\bar{R S} ≅ \bar{T Q}$	By CPCT

Step by step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Draw a line QS.

The figure after drawing the line $Q S$ is:

Step 3. Description of step.

It is given that $∠ R ≅ ∠ T$ and $\bar{R S} ∥ \bar{T Q}$ .

From the given figure it can be noticed that the angles $∠ R S Q$ and $∠ T Q S$ are the alternate interior angles.

Therefore, $∠ R S Q ≅ ∠ T Q S$ .

In the triangles $△ R S Q$ and $△ T Q S$ , the side is common.

Therefore, $Q S ≅ Q S$ by using the reflexive property.

Therefore, in the triangles $△ R S Q$ and $△ T Q S$ , it can be noticed that $∠ R ≅ ∠ T$ , $∠ R S Q ≅ ∠ T Q S$ and $Q S ≅ Q S$ ,

Therefore, the triangles $△ R S Q$ and $△ T Q S$ are congruent by using the AAS postulate.

Therefore, by corresponding parts of congruent triangles, it can be said that $\bar{R S} ≅ \bar{T Q}$ .

Step 4. Write the proof in two-column proof.

The proofs in two-column form is:

Statements	Reasons
Join $\bar{Q S}$	By construction
$∠ R ≅ ∠ T$	Given
$\bar{R S} ∥ \bar{T Q}$	Given
$∠ R S Q ≅ ∠ T Q S$	If a line intersects two parallel lines then each pair of alternate angles are equal.
$\bar{Q S} ≅ \bar{Q S}$	Reflexive property
$△ R S Q ≅ △ T Q S$	By AAS congruence
$\bar{R S} ≅ \bar{T Q}$	By CPCT

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Write the proofs in two-column form.
Given: $∠ R ≅ ∠ T$ ; $\bar{RS} ∥ \bar{QT}$
Prove: $\bar{RS} ≅ \bar{TQ}$
(Hint: What auxiliary line can you draw to form congruent triangles?)

Short Answer

Step by step solution

Step 1. Observe the given diagram.

Step 2. Draw a line QS.

Step 3. Description of step.

Step 4. Write the proof in two-column proof.

One App. One Place for Learning.

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Write the proofs in two-column form.Given: ∠R≅∠T;RS¯∥QT¯Prove:RS¯≅TQ¯(Hint: What auxiliary line can you draw to form congruent triangles?)

Short Answer

Step by step solution

Step 1. Observe the given diagram.

Step 2. Draw a line QS.

Step 3. Description of step.

Step 4. Write the proof in two-column proof.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Pure Maths

Applied Mathematics

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Write the proofs in two-column form.
Given: $∠ R ≅ ∠ T$ ; $\bar{RS} ∥ \bar{QT}$
Prove: $\bar{RS} ≅ \bar{TQ}$
(Hint: What auxiliary line can you draw to form congruent triangles?)