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)

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

you'll just earn points here

step-by-step explanation: