$6$ The figure shown can be used to prove the angles of a triangle theorem. $\newline$ Copy and complete: $\newline$ $\widehat{Q A C}=\ldots \ldots . \quad$ \{equal alternate angles $\}$ $\newline$ $\widehat{P A B}=$ $\newline$ \{equal alternate angles\} $\newline$ Now $\widehat{P A} B+\widehat{B A C}+\widehat{Q A C}=\ldots . . . \quad\{$ angles on a line $\}$ $\newline$ So, $\quad a+b+c=\ldots \ldots$ .

Full solution

6

The figure shown can be used to prove the angles of a triangle theorem.

\newline

Copy and complete:

\newline

\widehat{Q A C}=\ldots \ldots . \quad

\{equal alternate angles

\}

\newline

\widehat{P A B}=

\newline

\{equal alternate angles\}

\newline

Now

\widehat{P A} B+\widehat{B A C}+\widehat{Q A C}=\ldots . . . \quad\{

angles on a line

\}

\newline

So,

\quad a+b+c=\ldots \ldots

Identify Relationship: Identify the relationship between the angles given in the figure and the angles of a triangle theorem. $\newline$ The angles of a triangle theorem states that the sum of the interior angles of a triangle is $180$ degrees.
Find $QAC$ Value: Determine the value of $\widehat{QAC}$ using the given information. $\newline$ Since $\widehat{QAC}$ and $\widehat{PAB}$ are alternate angles and the lines are parallel, they are equal. Therefore, $\widehat{QAC} = a$ .
Find $PAB$ Value: Determine the value of $\widehat{PAB}$ using the given information. $\newline$ Similarly, $\widehat{PAB}$ is also an alternate angle to angle $b$ , so $\widehat{PAB} = b$ .
Calculate Sum: Calculate the sum of the angles on a straight line, which is known to be $180$ degrees. $\newline$ Now, $\widehat{PAB} + \widehat{BAC} + \widehat{QAC} = a + b + c$ , and since these angles form a straight line, their sum is $180$ degrees.
Conclude Total: Conclude the sum of the angles $a$ , $b$ , and $c$ . So, $a + b + c = 180$ degrees, which is the sum of the interior angles of a triangle.