Question

Please help me with two questions! Thank you so much!!

Answer 1

See below

Explanation:

DOK 3

`In\: \triangle ABC, \:\\</p><p>m\angle A + m\angle B + m\angle C = 180\degree \\\\</p><p>m\angle A + 76\degree + m\angle C = 180\degree \\\\</p><p>m\angle A + m\angle C = 180\degree - 76\degree \\\\</p><p>m\angle A + m\angle C = 104\degree \\\\</p><p>(1)/(2) (m\angle A + m\angle C) =(1)/(2)* 104\degree \\\\</p><p>\red{\bold{(1)/(2) (m\angle A + m\angle C) = 52\degree}}... (1)\\`

Since, AD and DC are bisectors of
`\angle A \: \&\: \angle C` respectively.

`m\angle DAC= (1)/(2)m\angle A... (2)\\\\</p><p>m\angle DCA= (1)/(2)m\angle C...(3) \\`

Adding equations (2) & (3), we find:

`m\angle DAC+m\angle DCA= (1)/(2)(m\angle A+m\angle C).... (4)\\`

From equations (1) & (4)

`m\angle DAC+m\angle DCA= 52\degree... (5)\\\\</p><p>In \triangle ADC, \\</p><p>m\angle ADC + m\angle DAC +m\angle DCA = 180\degree... (6)\\\\`

From equations (5) & (6), we find:

`m\angle ADC + 104\degree = 180\degree... (6)\\\\</p><p>m\angle ADC = 180\degree - 52\degree \\\\</p><p>\huge\purple {\boxed{m\angle ADC = 128\degree}} </p><p>`

DOK 4:

Let the measures of the angels 1, 2 and 3 be 4x, 5x and 6x respectively.

Therefore,

4x + 5x + 6x = 180°

15x = 180°

x = 180°/15

x = 12°

4x = 4*12° = 48°

5x = 5*12° = 60°

6x = 6*12° = 72°

`m\angle 1 = 48\degree \\\\</p><p>m\angle 2 = 60\degree \\\\</p><p>m\angle 3 = 72\degree \\`