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

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) 

Answer 2

Answer:

∠ABD = ∠ACD

Step-by-step 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.