Question

use the given information and the figure to complete the proof.
m∠fcb = 30°
m∠cbd = 120°
cf bisects ∠acb.
prove: △acb is an equilateral triangle. move the angles and angle measures to the lines to complete the proof.

1. since ∠cbd and ____ form a linear pair, their measures add up to __. therefore, m∠cba = ____.
2. since cf bisects ∠acb, ∠acf ≅ ____. therefore, m∠acf = __ and m∠acb = ____.
3. the triangle angle sum theorem means m∠acb + m∠cba + m∠bac = ____. therefore, m∠bac = ____.
4. since m∠cba = m∠acb = m∠bac, we know that △acb is equilateral.

∠bcf ∠caf ∠cba ∠ebd ∠edb
30° 60° 90° 120° 180°

Explanation:

Step1: Identify linear - pair angles

Since $\angle CBD$ and $\angle CBA$ form a linear pair, their measures add up to $180^{\circ}$. Given $m\angle CBD = 120^{\circ}$, then $m\angle CBA=180 - 120=60^{\circ}$.

Step2: Use angle - bisector property

Since $\overline{CF}$ bisects $\angle ACB$, $\angle ACF\cong\angle BCF$. Given $m\angle FCB = 30^{\circ}$, then $m\angle ACF = 30^{\circ}$ and $m\angle ACB=30 + 30 = 60^{\circ}$.

Step3: Apply Triangle Angle - Sum Theorem

The Triangle Angle - Sum Theorem means $m\angle ACB+m\angle CBA+m\angle BAC = 180^{\circ}$. Substitute $m\angle ACB = 60^{\circ}$ and $m\angle CBA = 60^{\circ}$ into the formula: $60+60+m\angle BAC = 180$. Solving for $m\angle BAC$, we get $m\angle BAC=180-(60 + 60)=60^{\circ}$.

Answer:

1. $\angle CBA$, $180^{\circ}$, $60^{\circ}$
2. $\angle BCF$, $30^{\circ}$, $60^{\circ}$
3. $180^{\circ}$, $60^{\circ}$