Question

A kite is a quadrilateral with two pairs of adjacent, congruent sides. Prove the two angles between the non-congruent sides are congruent. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted.

Answer 1

∠ABD = ∠ACD

Explanation:

Given : ABCD is a quadrilateral

AB≈AC

DB≈DC

To Prove : ∠ABD = ∠ACD

Proof : We know that AB≈AC and DB≈DC

Draw a line join A and D (refer attached figure)

This divides quadrilateral into two triangles named as ΔABD and ΔACD

Now , We know that

AB≈AC (given) this means AB = AC

DB≈DC(given) this means DB=DC

AD=AD(common)

So , by SSS property of congruency ΔABD and ΔACD are congruent .

( SSS property is when all three sides in one triangle are the same length as the corresponding sides in the other)

Hence, ΔABD ≅ ΔACD

Since ΔABD ≅ ΔACD (Proved above)

⇒∠ABD = ∠ACD

Thus , Two angles between non-congruent sides are congruent.

Answer 2

let the kite is the quadrilateral ABCD with the two pairs of adjacent congruent sides. (the diagram is also attached)

Now, its given in question, see from figure,
AB is congruent to AD
BC is congruent to DC

Now, let us join the points A & C to form AC ; and points B and D to form BD.
So, AC is common side to triangles ABC and ADC.
So, Because AB ≈AD
BC ≈ DC
And, AC is common, therefore,
triangle ABC is congruent to triangle ADC

⇒∠ ABC ≈ ∠ADC (These are the angles between the non-congruent sides)