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[Audio] Let's continue our topic about Proving theorems on the different kinds of Parallelogram..

[Audio] Now, let us proceed in RHOMBUS. RHOMBUS is a parallelogram with all four sides congruent. Here are the PROPERTIES OF RHOMBUS All sides are congruent Opposite sides are parallel Opposite angles are congruent Consecutive angles are supplementary Each diagonal separates the rhombus into two congruent triangles Diagonals bisect each other and are perpendicular And each diagonal bisects a pair of opposite angles.

[Audio] Theorem number 3: The diagonal of a rhombus are perpendicular. Given: Rhombus R O S E. This is the figure of rhombus R O S E with a midpoint H. Now, let us prove that line segment R S is perpendicular to line segment O E. Proof: Here are the statements and the reasons. Number 1: Rhombus R O S E. Reason: It is given. Number 2: Line segment O S is congruent to line segment R O. By: Definition of rhombus. Line segment R H is congruent to line segment H S and Line segment E H is congruent to line segment H O. Reason: The diagonals of a parallelogram bisects each other. Number 4: H is the midpoint of line segment R S. Reason: Line segment E O bisect line segment R S at H..

[Audio] Number 5: Line segment R H is congruent to line segment H S. Reason: Definition of midpoint. Number 6: Line segment O H is congruent to line segment O H. Reason: Reflexive property. Number 7: Triangle R H O is congruent to triangle S H O. By: SSS or side side side congruence postulate. Number 8: Angle R H O is congruent to angle S H O. Reason: Corresponding Parts of Congruent Triangles are Congruent ( CPCTC). Number 9: Angle R H O and angle S H O are right angles. Reason: Angle R H O and angle S H O are linear pair and congruent. Number 10: Line segment R S is perpendicular to line segment O E. Reason: Perpendicular lines meet to form right angles. Therefore, we conclude that line segment R S is perpendicular to line segment O E..

[Audio] Let us discuss another theorem on rhombus. Theorem number 4: Each diagonal of a rhombus bisect opposite angles. Given: Rhombus V W X Y. This is the figure of rhombus V W X Y. Now, let us Prove that Angle1 is congruent to angle 2 and angle 3 is congruent to angle 4. Proof: Here are the statements and the reasons. Number 1: Rhombus V W X Y Reason: It is given. Number 2: line segment Y V is congruent to line segment V W and line segment W X is congruent to line segment X Y. By: Definition of rhombus. Number 3: Line segment W Y is congruent to Y W. By: Reflexive property. Number 4: Triangle Y V W is congruent to triangle W X Y. Reason: SSS or side side side Congruence Postulate. Number 5: Angle 1 is congruent to angle 2 and angle 3 is congruent to angle 4. By: Corresponding Parts of Congruent Triangle are congruent ( CPCTC). Therefore, we can say that Angle 1 is congruent to angle 2 and angle 3 is congruent to angle 4..

[Audio] Let us recall the definition of a square. SQUARE is a rectangle with four sides are congruent. Here are the PROPERTIES OF SQUARE All sides are congruent All angles are congruent Diagonals bisect each other Diagonals are perpendicular Diagonals are congruent Consecutive angles are supplementary Each diagonal separates the square into two congruent triangles.

. R E S O. Proof:. STATEMENTS. REASONS. 1. ROSE is a square.

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