How to prove that a quadrilateral is a rhombus in two ways: by geometric construction and by deductive reasoning.

[color=#1155cc]1. A [b]rhombus[/b] is a quadrilateral with sides of equal length.[br][br]2. [b]Geometric construction[/b] is the process of creating geometric shapes using only a compass and a straight edge (ruler) without numerical measurements. [br][br]3. [b]Proving by geometric construction[/b] involves demonstrating the truth of a geometric statement using a compass and straightedge. These constructions rely on logical deductions and established geometric principles. [br][br]4. [b]Proving by deductive reasoning[/b] involves proving geometric statements using a logical sequence of steps based on previously accepted facts, definitions, postulates, and theorems[/color]

[color=#9900ff]Two congruent circles with centers A and B intersect at points D and C. Prove that ACBD is a rhombus.[/color]

The construction uses the point tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_complexnumber.png[/icon], the line segment tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon], and the circle through center and point tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_circle2.png[/icon] which is the same as the compass tool [icon]/images/ggb/toolbar/mode_compasses.png[/icon]to construct your rhombus ACBD.

Answer these problems and enhance your skill in proving by deductive reasoning in statement-reason form.

[color=#9900ff]Given a quadrilateral PQRS. PQ and PR are radii of Circle P, and RQ and RS are radii of Circle R. If Circle P and Circle R are congruent, prove that quadrilateral PQRS is a [b]rhombus[/b].[/color]

[color=#9900ff]Refer to Problem 2. If Circle P and Circle R are not congruent, prove that quadrilateral PQRS is a [b]kite[/b].[/color]