Q28 Given: 聽鈻盇BZY; 聽ZY炉鈮匓X炉;... [FREE SOLUTION]

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 5: Q28 (page 188)

Given: $鈻� ABZY$ ; $\overset{炉}{ZY} 鈮� \overset{炉}{BX}$ ; $鈭� 1 鈮� 鈭� 2$
Prove: ABZY is a rhombus.

Expert verified

The two-column proof of $ABZY$ is a rhombus is:

Statement	Reason
$\overset{炉}{ZY} 鈮� \overset{炉}{BX}$	Given.
$鈭� 1 鈮� 鈭� 2$	Given.
$鈭� AXB 鈮� 鈭� YXZ$	Vertical angles are congruent.
$螖 AXB 鈮� 螖 YXZ$	AAS.
$AX 鈮� XZ$	CPCTC.
$BX 鈮� XY$	CPCTC.
Hence $鈭� YAB$ is a right angle.	Definition of a parallelogram.
$\overset{炉}{AZ}$ is perpendicular bisector to $\overset{炉}{BY}$ and $\overset{炉}{BY}$ is perpendicular bisector to $\overset{炉}{A Z}$ .	Definition of perpendicular bisector.
$A B Z Y$ is a rhombus.	Definition of a rhombus.

Parallelogram $A B Z Y$ is:

Here, $\overset{炉}{Z Y} 鈮� \overset{炉}{B X}$ and $鈭� 1 鈮� 鈭� 2$ .

Statement	Reason
$\overset{炉}{Z Y} 鈮� \overset{炉}{B X}$	Given.
$鈭� 1 鈮� 鈭� 2$	Given.
$鈭� A X B 鈮� 鈭� Y X Z$	Vertical angles are congruent.
$螖 A X B 鈮� 螖 Y X Z$	AAS.
$A X 鈮� X Z$	CPCTC.
$B X 鈮� X Y$	CPCTC.
Hence $鈭� Y A B$ is a right angle.	Definition of a parallelogram.
$\overset{炉}{A Z}$ is perpendicular bisector to $\overset{炉}{B Y}$ and $\overset{炉}{B Y}$ is perpendicular bisector to $\overset{炉}{A Z}$ .	Definition of perpendicular bisector.
$A B Z Y$ is a rhombus.	Definition of a rhombus.