Question

in △abc, line segments ac ≅ bc and m∠bca = 70°. 15. what is the measure of ∠abc? a. 20° b. 35° c. 55° d. 70°

Explanation:

Step1: Recall isosceles - triangle property

Since $AC\cong BC$, $\triangle ABC$ is an isosceles triangle. In an isosceles triangle, the base - angles are equal. Let $\angle BAC=\angle ABC = x$.

Step2: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So, in $\triangle ABC$, we have $\angle BCA+\angle BAC+\angle ABC = 180^{\circ}$. Given $\angle BCA = 70^{\circ}$, and $\angle BAC=\angle ABC=x$, we can write the equation as $70^{\circ}+x + x=180^{\circ}$.

Step3: Solve the equation for $x$

Combining like terms, we get $70^{\circ}+2x = 180^{\circ}$. Subtract $70^{\circ}$ from both sides: $2x=180^{\circ}-70^{\circ}=110^{\circ}$. Then divide both sides by 2: $x = 55^{\circ}$.

Answer:

C. $55^{\circ}$