USING THE IMAGE ATTACHED BELOW!
formula, shapes, kite, degrees

The diagram shows a quadrilateral ABCD with each of its sides extended

AB=AD

Show that ABCD is a kite

Give a reason for each stage of your working.

Step-by-step explanation:

As given, AB = AD, so that triangle ABD is an isosceles triangle.

=> In triangle ABD: m∠ABD = m∠ADB

As we can see:

+) The measure of supplementary angle of Angle ADC is 65°.

=> m∠ADC = 180° - 65° = 115°.

+) The measure of vertical angle of Angle BCD is 20°.

=>  m∠BCD = 20°.

+) m∠BAD = 110°

As ABCD is a quadrilateral, so that total measure of its 4 interior angles are 360°:

⇒ m∠ADC + m∠BCD + m∠CBA + m∠BAD = 360°

⇔ 115° + 20° + m∠CBA + 110° = 360°

⇔ m∠CBA = 360° - 110° - 20° - 115° = 115°

⇒ m∠CBA = m∠ADC = 115°

We have:

+) m∠CBA = m∠CBD + m∠ABD

+) m∠ADC = m∠CDB + m∠ADB

As m∠CBA = m∠ADC; m∠ABD = m∠ADB so that: m∠CBD = m∠CDB

In triangle BDC, m∠CBD = m∠CDB

=> BDC is an  isosceles triangle.

=> CD = CB

In quadrilateral, CD = CB; AB = AD

=> there are two disjoint pairs of consecutive sides of a quadrilateral are congruent

=> ABCD is a kite (reverse of the kite definition)