Questions: If the vector A B is bisecting A D C A D, find the following: x= m angle C A B= m angle B A D= m angle C A D=

Transcript text: If $\overrightarrow{A B}$ is bisecting $A D C A D$, find the following: $x=$ $\qquad$ $m \angle C A B=$ $\qquad$ $m \angle B A D=$ $\qquad$ $m \angle C A D=$ $\qquad$

Solution

Solution Steps

Step 1: Set up the equation for the bisected angles

Since $ \overline{AB} $ is bisecting $ \angle CAD $, the measure of $ \angle CAB $ is equal to the measure of $ \angle BAD $. Therefore, we can set up the equation: \[ 6x - 2 = 4x + 8 \]

Step 2: Solve for $ x $

To find $ x $, we solve the equation: \[ 6x - 2 = 4x + 8 \] Subtract $ 4x $ from both sides: \[ 2x - 2 = 8 \] Add 2 to both sides: \[ 2x = 10 \] Divide by 2: \[ x = 5 \]

Step 3: Calculate $ m\angle CAB $

Substitute $ x = 5 $ into the expression for $ m\angle CAB $: \[ m\angle CAB = 6x - 2 = 6(5) - 2 = 30 - 2 = 28^\circ \]

Step 4: Calculate $ m\angle BAD $

Since $ \overline{AB} $ bisects $ \angle CAD $, $ m\angle BAD $ is equal to $ m\angle CAB $: \[ m\angle BAD = 28^\circ \]

Step 5: Calculate $ m\angle CAD $

Since $ \angle CAD $ is the sum of $ \angle CAB $ and $ \angle BAD $: \[ m\angle CAD = m\angle CAB + m\angle BAD = 28^\circ + 28^\circ = 56^\circ \]