Chapter 4: Q6. (page 131)

Prove the converse of the statement in Exercise 5: If both pairs of opposite sides of a quadrilateral are congruent, then they are also parallel.
Given: $\bar{S K} ≅ \bar{N R}; \bar{S N} ≅ \bar{K R}$
Prove: $\bar{S K} ∥ \bar{N R}; \bar{S N} ∥ \bar{K R}$

Short Answer

Expert verified

Statement	Reason
1. $\bar{S K} ≅ \bar{N R}$	Given
2. $\bar{S R} ≅ \bar{S R}$	Common line segment
3. $\bar{S N} ≅ \bar{K R}$	Given
4. $Δ S N R ≅ Δ R K S$	SSS congruency criteria
5. $∠ 1 ≅ ∠ 3; ∠ 2 ≅ ∠ 4$	corresponding parts of congruent triangle are congruent
6. $\bar{S K} ∥ \bar{N R}; \bar{S N} ∥ \bar{K R}$	When alternate interior angles are congruent then lines are parallel

Step by step solution

Step 1. Show that ΔSNR≅ΔRKS.

Since $\bar{S R} ≅ \bar{S R}$ , as it is common line in both triangles

Also, using given

Thus, $Δ S N R ≅ Δ R K S$ by SSS (Side-Side-Side) congruency criteria

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Since, corresponding parts of congruent triangle are congruent

Thus, $∠ 1 ≅ ∠ 3; ∠ 2 ≅ ∠ 4$

Step 3. Show that SK¯∥NR¯; SN¯∥KR¯.

From figure, $(∠ 1 - ∠ 3)$ and $(∠ 2 - ∠ 4)$ forms pairs of alternate interior angles

When transversal line intersect two lines such that alternate interior angles are congruent then two lines are parallel.

Thus, $\bar{S K} ∥ \bar{N R}; \bar{S N} ∥ \bar{K R}$ 1

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Prove the converse of the statement in Exercise 5: If both pairs of opposite sides of a quadrilateral are congruent, then they are also parallel.
Given: $\bar{S K} ≅ \bar{N R}; \bar{S N} ≅ \bar{K R}$
Prove: $\bar{S K} ∥ \bar{N R}; \bar{S N} ∥ \bar{K R}$

Short Answer

Step by step solution

Step 1. Show that ΔSNR≅ΔRKS.

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Step 3. Show that SK¯∥NR¯; SN¯∥KR¯.

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Prove the converse of the statement in Exercise 5: If both pairs of opposite sides of a quadrilateral are congruent, then they are also parallel.Given:SK¯≅NR¯;SN¯≅KR¯Prove:SK¯∥NR¯;SN¯∥KR¯

Short Answer

Step by step solution

Step 1. Show that ΔSNR≅ΔRKS.

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Step 3. Show that SK¯∥NR¯; SN¯∥KR¯.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Prove the converse of the statement in Exercise 5: If both pairs of opposite sides of a quadrilateral are congruent, then they are also parallel.
Given: $\bar{S K} ≅ \bar{N R}; \bar{S N} ≅ \bar{K R}$
Prove: $\bar{S K} ∥ \bar{N R}; \bar{S N} ∥ \bar{K R}$