Learn to Prove: Kite

Object of Learning

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

Definition

[color=#1155cc]1. A [b]kite[/b] is a quadrilateral with two pairs of adjacent sides having equal lengths. [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]

Problem 1

[color=#9900ff]Two circles with centers A and C intersect at D and E. Prove that ACDE is a kite.[/color]

Proof by Geometric Construction

The video below shows how to construct a kite.[br]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 compass tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_compasses.png[/icon]to construct the kite ADCE.

Practice how to construct a kite.

Proving by deductive reasoning

The series of questions below will give you hints on how to construct the deductive proof. Refer to the figure shown in the video clip, as you may have used different names for the vertices of the kite in your construction.

Is the statement AD=AE true or false?

False True Neither true or false

Which of the following guarantees that AD=AE?

They are radii of Circle A, so they must have equal lengths. They are radii of Circle C, so they must have equal lengths.

Is this statement CD=CE true or false?

False True Neither true or false

Which of the following justifies that CD=CE?

They are radii of Circle A, so they must have equal lengths. They are radii of Circle C, so they must have equal lengths.

Is it enough to conclude that AEBD is a kite?

Yes Hard to say No

Which of the following statements justifies the conclusion that AECD is a kite?

The shape AECD is a kite based on the definition of a kite. The shape AECD has two pairs of equal sides: AE=DC and AD=CE. The shape AECD is a quadrilateral The shape AECD looks like a kite. The shape AECD has two pairs of equal sides: AE=AD and CD=CE.

Problem 2

[color=#9900ff]Circle M and Circle N intersect at points O and T. Prove that quadrilateral MONT is a kite by deductive reasoning.[/color]

Statement-Reason form of Deductive Proof

Deductive reasoning can be written in statement-reason form. Solve problem 2 by supplying the reason for each statement or claim in the following table.