Question

4 m∠bca = __________ m∠abc = __________

Explanation:

Step1: Use linear - angle property

The angle $\angle DAB = 97^{\circ}$, and $\angle BAC$ and $\angle DAB$ are linear - pair angles. So, $\angle BAC=180 - 97=83^{\circ}$.

Step2: Apply angle - sum property of a triangle

In $\triangle ABC$, the sum of interior angles is $180^{\circ}$. So, $\angle BAC+\angle ABC+\angle BCA = 180^{\circ}$. Substitute $\angle BAC = 83^{\circ}$, $\angle ABC=(14x - 1)^{\circ}$, and $\angle BCA=(2x + 2)^{\circ}$ into the equation:

$$83+(14x - 1)+(2x + 2)=180$$

$$83 + 14x-1+2x + 2=180$$

$$84+16x=180$$

$$16x=180 - 84$$

$$16x=96$$

$$x = 6$$

Step3: Find $\angle BCA$

Substitute $x = 6$ into the expression for $\angle BCA$: $\angle BCA=(2x + 2)^{\circ}$. Then $\angle BCA=(2\times6 + 2)^{\circ}=14^{\circ}$.

Step4: Find $\angle ABC$

Substitute $x = 6$ into the expression for $\angle ABC$: $\angle ABC=(14x - 1)^{\circ}$. Then $\angle ABC=(14\times6 - 1)^{\circ}=83^{\circ}$.

Answer:

$m\angle BCA = 14^{\circ}$
$m\angle ABC = 83^{\circ}$