Q6. Prove the converse of the statem... [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: 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: $\overset{炉}{S K} 鈮� \overset{炉}{N R}; \overset{炉}{S N} 鈮� \overset{炉}{K R}$
Prove: $\overset{炉}{S K} 鈭� \overset{炉}{N R}; \overset{炉}{S N} 鈭� \overset{炉}{K R}$

Short Answer

Expert verified

Statement	Reason
1. $\overset{炉}{S K} 鈮� \overset{炉}{N R}$	Given
2. $\overset{炉}{S R} 鈮� \overset{炉}{S R}$	Common line segment
3. $\overset{炉}{S N} 鈮� \overset{炉}{K R}$	Given
4. $螖 S N R 鈮� 螖 R K S$	SSS congruency criteria
5. $鈭� 1 鈮� 鈭� 3; 鈭� 2 鈮� 鈭� 4$	corresponding parts of congruent triangle are congruent
6. $\overset{炉}{S K} 鈭� \overset{炉}{N R}; \overset{炉}{S N} 鈭� \overset{炉}{K R}$	When alternate interior angles are congruent then lines are parallel

Step by step solution

Step 1. Show that ΔSNR≅ΔRKS.

Since $\overset{炉}{S R} 鈮� \overset{炉}{S R}$ , as it is common line in both triangles

Also, using given

Thus, $螖 S N R 鈮� 螖 R K S$ by SSS (Side-Side-Side) congruency criteria

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Since, corresponding parts of congruent triangle are congruent

Thus, $鈭� 1 鈮� 鈭� 3; 鈭� 2 鈮� 鈭� 4$

Step 3. Show that SK¯∥NR¯; SN¯∥KR¯.

From figure, $(鈭� 1 鈭� 鈭� 3)$ and $(鈭� 2 鈭� 鈭� 4)$ 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, $\overset{炉}{S K} 鈭� \overset{炉}{N R}; \overset{炉}{S N} 鈭� \overset{炉}{K R}$ 1

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91影视

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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91影视

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炉鈮�NR炉;SN炉鈮�KR炉Prove:SK炉鈭�NR炉;SN炉鈭�KR炉

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

Logic and Functions

Mechanics Maths

Discrete Mathematics

Statistics

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.