Question

given $\triangle abc$ with $ab = ac$. use deductive reasoning to complete the following sentence. if $\angle b = 40^{\circ}$, then ______. $\bigcirc$ $m\angle c = 100^{\circ}$ $\bigcirc$ $m\angle a = 100^{\circ}$ $\bigcirc$ $m\angle a = 40^{\circ}$ $\bigcirc$ $m\angle c = 80^{\circ}$

Explanation:

Step1: Identify triangle type

Since \( AB = AC \), \( \triangle ABC \) is isosceles with \( \angle B=\angle C \).

Step2: Find \( \angle C \)

Given \( \angle B = 40^\circ \), so \( \angle C=\angle B = 40^\circ \).

Step3: Calculate \( \angle A \)

Using triangle angle - sum property (\( \angle A+\angle B+\angle C = 180^\circ \)):
\( \angle A=180^\circ-\angle B - \angle C \)
Substitute \( \angle B = 40^\circ \) and \( \angle C = 40^\circ \):
\( \angle A=180^\circ - 40^\circ-40^\circ=100^\circ \)

Answer:

\( m\angle A = 100^\circ \) (the option corresponding to this is the one with \( m\angle A = 100^\circ \))