Q5. Prove the following statement: I... [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: Q5. (page 131)

Prove the following statement: If both pairs of opposite sides of a quadrilateral are parallel, then they are also congruent.
Given: $\overset{炉}{S K} 鈭� \overset{炉}{N R}; \overset{炉}{S N} 鈭� \overset{炉}{K R}$
Prove: $\overset{炉}{S K} 鈮� \overset{炉}{N R}; \overset{炉}{S N} 鈮� \overset{炉}{K R}$

Short Answer

Expert verified

Statement	Reason
1. $鈭� 1 鈮� 鈭� 3$	Alternate interior angles
2. $\overset{炉}{S R} 鈮� \overset{炉}{S R}$	Common line segment
3. $鈭� 4 鈮� 鈭� 2$	Alternate interior angles
4. $螖 S N R 鈮� 螖 R K S$	ASA congruency criteria
5. $\overset{炉}{S K} 鈮� \overset{炉}{N R}; \overset{炉}{S N} 鈮� \overset{炉}{K R}$	corresponding parts of congruent triangle are congruent

Step by step solution

Step 1. Observe from figure.

$\overset{炉}{S R}$ is transversal to the parallel lines $\overset{炉}{S K} 鈭� \overset{炉}{N R}; \overset{炉}{S N} 鈭� \overset{炉}{K R}$

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Since $(鈭� 1 鈭� 鈭� 3)$ and $(鈭� 2 鈭� 鈭� 4)$ forms pairs of alternate interior angles

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

Thus, $鈭� 1 鈮� 鈭� 3; 鈭� 2 鈮� 鈭� 4$

Step 3. Show that ΔSNR≅ΔRKS.

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

Also, $鈭� 1 鈮� 鈭� 3; 鈭� 2 鈮� 鈭� 4$

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

Step 4. Show that SK¯≅NR¯; SN¯≅KR¯.

Since, corresponding parts of congruent triangle are congruent

Thus, $\overset{炉}{S K} 鈮� \overset{炉}{N R}; \overset{炉}{S N} 鈮� \overset{炉}{K R}$

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Short Answer

Step by step solution

Step 1. Observe from figure.

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Step 3. Show that ΔSNR≅ΔRKS.

Step 4. Show that SK¯≅NR¯; SN¯≅KR¯.

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91影视

Prove the following statement: If both pairs of opposite sides of a quadrilateral are parallel, then they are also congruent.Given:SK炉鈭�NR炉;SN炉鈭�KR炉Prove:SK炉鈮�NR炉;SN炉鈮�KR炉

Short Answer

Step by step solution

Step 1. Observe from figure.

Step 2. Show that ∠1≅∠3; ∠2≅∠4.

Step 3. Show that ΔSNR≅ΔRKS.

Step 4. Show that SK¯≅NR¯; SN¯≅KR¯.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Statistics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.