Question

А
Triangles ACD and BCD are isosceles. Angle BAC has a
measure of 33 degrees and angle BDC has a measure of 35
degrees. Find the measure of angle ABD.
B

Answer 1

Answer:

The measure of angle ∠ABD is 125°

Step-by-step explanation:

The given parameters are;

Triangle of ΔACD = Isosceles triangle

Triangle of ΔBCD = Isosceles triangle

m∠BAC = 33°

m∠BDC = 35°

Whereby ΔBCD and ΔACD have their vertex in the same direction, we can have;

$\overline {AD}$ ≅ $\overline {AC}$ Given sides legs of triangle ACD

$\overline {BD}$ ≅ $\overline {BC}$ Given sides legs of triangle BCD

$\overline {BA}$ ≅ $\overline {BA}$ Reflexive property

∴ ΔDAB ≅ ΔCAB  by SSS (Side-Side-Side) congruency rule

m∠BAC ≅ m∠BAD CPCTC (Congruent Parts of Congruent Triangles are Congruent)

∴ m∠BAC ≅ m∠BAD = 33° Definition of congruency

m∠BAC + m∠BAD = 33° + 33° = 66° = m∠DAC Angle addition postulate

m∠ADC = m∠ACD = (180 - m∠DAC)/2 = (180 - 66)/2 = 57° Base angles of an isosceles triangle

∴ m∠ADC = m∠ACD = 57°

m∠BDC = 35° = m∠BCD Base angles of isosceles triangle ΔBCD

m∠BCD + m∠DBC + m∠BCD = 180° Sum of the interior angles of a triangle theorem

m∠DBC = 180° - (m∠BCD +  m∠BCD) = 180° - (35° +  35°) = 110°

m∠DBC = 110°

∴ m∠ABD ≅ m∠ABC CPCTC

m∠ABD = m∠ABC Definition of congruency

m∠DBC + m∠ABD + m∠ABC = 360° Sum of angles at a point

m∠DBC + m∠ABD + m∠ABD = 360°

m∠DBC + 2 × m∠ABD = 360°

110° + 2 × m∠ABD = 360°

2 × m∠ABD = 360° - 110° = 250°

2 × m∠ABD = 250°

m∠ABD = 250°/2 = 125°

m∠ABD = 125°.

Answer 2

The measure of angle ABD is equal to $125^\circ$ and this can be determined by using the properties of the triangle.

Given :

• Triangles ACD and BCD are isosceles.
• Angle BAC has the  measure of 33 degrees and angle BDC has a measure of 35  degrees.

Given that side $\rm AD \cong AC$.

Given that side $\rm BD \cong BC$.

$\rm BA \cong BA$ according to the reflexive property.

So, according to the above statements triangle DAB is congruent to triangle CAB by the SSS postulate.

Now according to CPCTC Angle BAD is congruent to angle BAC.

$\rm \angle BAD + \angle BAC = 33 + 33 = 66^\circ$

$\rm \angle DAC = 66^\circ$   (according to the angle addition postulate)

$\rm \angle ADC = \angle ACD = \dfrac{180-\angle DAC }{2} = \dfrac{180-66}{2}=57^\circ$

$\rm \angle BDC = \angle BCD = 35^\circ$

Now, according to the sum of the interior angle of the triangle theorem:

$\rm \angle BCD+\angle DBC+\angle BDC = 180^\circ$

$\rm \angle DBC = 180 - 35-35 = 110^\circ$

Therefore, the angle ABD and angle ABC are congruent according to the CPCTC.

Now, according to the definition of congruency angle ABD is equal to angle ABC.

Now, again according to the sum of angles at a point:

$\rm \angle DBC + \angle ABD + \angle ABC = 360^\circ$

$\rm \angle DBC + \angle ABD + \angle ABD = 360^\circ$

$\rm \angle DBC + 2\times \angle ABD = 360^\circ$

$\rm 110+2\times \angle ABD = 180^\circ$

$\rm 2\times \angle ABD = 360-110$

$\rm 2\times \angle ABD = 250^\circ$

$\rm \angle ABD = 125^\circ$

For more information, refer to the link given below:

https://brainly.com/question/24580745