Question

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

Answer 1

The measures of angle ABD is  mathematically given as

$\angle \mathrm{ABD}=125^{\circ}$

This is further explained below.

What is the measure of angle ABD?

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

$\angle \mathrm{BAD}+\angle \mathrm{BAC}=33+33=66^{\circ}\\\\\angle \mathrm{DAC}=66^{\circ} \\\\\angle \mathrm{ADC}=\angle \mathrm{ACD}=\frac{180-\angle \mathrm{DAC}}\\\\\\2}=\frac{180-66}{2}=57^{\circ}\\\\\angle \mathrm{BDC}=\angle \mathrm{BCD}=35^{\circ}$

Now, in accordance with the theory about the sum of the inner angles of the triangle:

$\angle \mathrm{BCD}+\angle \mathrm{DBC}+\angle \mathrm{BDC}=180^{\circ}\\\\ \angle \mathrm{DBC}=180-35-35=110^{\circ}$

As a result, the angle ABC and the angle ABD have the same congruence when viewed in light of the CPCTC.

Now, according to the notion of congruency, the angle ABD is equivalent to the angle ABC.

To continue, in accordance with the total of the angles at a point:

$\begin{aligned}&\angle \mathrm{DBC}+\angle \mathrm{ABD}+\angle \mathrm{ABC}=360^{\circ} \\&110+2 \times \angle \mathrm{ABD}=180^{\circ} \\&2 \times \angle \mathrm{ABD}=360-110 \\&2 \times \angle \mathrm{ABD}=250^{\circ} \\&\angle \mathrm{ABD}=125^{\circ}\end{aligned}$

In conclusion, the measure of angle ABD

$\angle \mathrm{ABD}=125^{\circ}$

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