Question

in the given figure, find ∠cbd. also, if ∠a = 75°, ∠c = 55°, find ∠abd

Explanation:

Step1: Recall angle - sum property of a triangle

In $\triangle ABD$, we know that the sum of interior angles of a triangle is $180^{\circ}$. Given $\angle A = 75^{\circ}$ and $\angle ADB=55^{\circ}$. Let $\angle ABD = x$. Then $x + 75^{\circ}+55^{\circ}=180^{\circ}$.

Step2: Solve for $\angle ABD$

$x=180^{\circ}-(75^{\circ} + 55^{\circ})=180^{\circ}-130^{\circ}=50^{\circ}$. So $\angle ABD = 50^{\circ}$.

Step3: Find $\angle CBD$

We are not given any information about the relationship between $\angle ABD$ and $\angle CBD$ other than they are angles at point $B$. Assuming $\angle ABC$ is a straight - line angle ($180^{\circ}$) and $\angle ABD$ and $\angle CBD$ are adjacent angles that make up $\angle ABC$. If we assume $\angle ABC = 180^{\circ}$, and we know $\angle ABD = 50^{\circ}$, then $\angle CBD=180^{\circ}-\angle ABD$.

Step4: Calculate $\angle CBD$

$\angle CBD = 180^{\circ}-50^{\circ}=130^{\circ}$.

Answer:

$\angle ABD = 50^{\circ}$, $\angle CBD = 130^{\circ}$