Since QP and QB are equal the triangle PQB the angles:
![\begin{gathered} \measuredangle QPB=\measuredangle QBP \\ \measuredangle QPB=x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20QPB%3D%5Cmeasuredangle%20QBP%20%5C%5C%20%5Cmeasuredangle%20QPB%3Dx%20%5Cend%7Bgathered%7D)
The last angle can be found by adding all the internal angles and making it equal to 180 degrees.
![\begin{gathered} \measuredangle QPB+\measuredangle QBP+\measuredangle BQP=180 \\ x+x+\measuredangle BQP=180 \\ \measuredangle BQP=180-2x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20QPB%2B%5Cmeasuredangle%20QBP%2B%5Cmeasuredangle%20BQP%3D180%20%5C%5C%20x%2Bx%2B%5Cmeasuredangle%20BQP%3D180%20%5C%5C%20%5Cmeasuredangle%20BQP%3D180-2x%20%5Cend%7Bgathered%7D)
The angle BQP and the angle AQP are suplementary, this means that their sum is equal to 180 degrees. So we have:
![\begin{gathered} \measuredangle AQP+\measuredangle BQP=180 \\ \measuredangle AQP+180-2x=180 \\ \measuredangle AQP=180-180+2x \\ \measuredangle AQP=2x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20AQP%2B%5Cmeasuredangle%20BQP%3D180%20%5C%5C%20%5Cmeasuredangle%20AQP%2B180-2x%3D180%20%5C%5C%20%5Cmeasuredangle%20AQP%3D180-180%2B2x%20%5C%5C%20%5Cmeasuredangle%20AQP%3D2x%20%5Cend%7Bgathered%7D)
Since the sides AP and PQ are equal, then the angle PAQ is equal to AQP.
![\begin{gathered} \measuredangle PAQ=\measuredangle AQP \\ \measuredangle PAQ=2x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20PAQ%3D%5Cmeasuredangle%20AQP%20%5C%5C%20%5Cmeasuredangle%20PAQ%3D2x%20%5Cend%7Bgathered%7D)
To find the last angle on that triangle we can add all the internal angles and make it to 180 degrees.
![\begin{gathered} \measuredangle PAQ+\measuredangle AQP+\measuredangle APQ=180 \\ 2x+2x+\measuredangle APQ=180 \\ \measuredangle APQ=180-4x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20PAQ%2B%5Cmeasuredangle%20AQP%2B%5Cmeasuredangle%20APQ%3D180%20%5C%5C%202x%2B2x%2B%5Cmeasuredangle%20APQ%3D180%20%5C%5C%20%5Cmeasuredangle%20APQ%3D180-4x%20%5Cend%7Bgathered%7D)
The angle APC is suplementary with the sum of the angles APQ and BPQ. So we have:
![\begin{gathered} \measuredangle APC+\measuredangle APQ+\measuredangle BPQ=180 \\ \measuredangle APC+180-4x+x=180 \\ \measuredangle APC=3x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20APC%2B%5Cmeasuredangle%20APQ%2B%5Cmeasuredangle%20BPQ%3D180%20%5C%5C%20%5Cmeasuredangle%20APC%2B180-4x%2Bx%3D180%20%5C%5C%20%5Cmeasuredangle%20APC%3D3x%20%5Cend%7Bgathered%7D)
The sides AP and AC are equal, therefore the angles APC and ACP are also equal.
![\begin{gathered} \measuredangle ACP=\measuredangle APC \\ \measuredangle ACP=3x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20ACP%3D%5Cmeasuredangle%20APC%20%5C%5C%20%5Cmeasuredangle%20ACP%3D3x%20%5Cend%7Bgathered%7D)
Then we can find the last angle on that triangle.
![\begin{gathered} \measuredangle CAP+\measuredangle ACP+\measuredangle APC=180 \\ \measuredangle CAP+3x+3x=180 \\ \measuredangle CAP=180-6x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20CAP%2B%5Cmeasuredangle%20ACP%2B%5Cmeasuredangle%20APC%3D180%20%5C%5C%20%5Cmeasuredangle%20CAP%2B3x%2B3x%3D180%20%5C%5C%20%5Cmeasuredangle%20CAP%3D180-6x%20%5Cend%7Bgathered%7D)
The angle CAB is equal to the sum of CAP and PAQ. So we have:
![\begin{gathered} \measuredangle CAB=\measuredangle CAP+\measuredangle PAQ \\ \measuredangle CAB=180-6x+2x \\ \measuredangle CAB=180-4x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20CAB%3D%5Cmeasuredangle%20CAP%2B%5Cmeasuredangle%20PAQ%20%5C%5C%20%5Cmeasuredangle%20CAB%3D180-6x%2B2x%20%5C%5C%20%5Cmeasuredangle%20CAB%3D180-4x%20%5Cend%7Bgathered%7D)
Since the sides AB and BC are equal, then the angles ACB and CAB must also be equal. We can find the value of x with this.
![\begin{gathered} \measuredangle BAC=\measuredangle ACB \\ 180-4x=3x \\ 7x=180 \\ x=\frac{180}{7} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cmeasuredangle%20BAC%3D%5Cmeasuredangle%20ACB%20%5C%5C%20180-4x%3D3x%20%5C%5C%207x%3D180%20%5C%5C%20x%3D%5Cfrac%7B180%7D%7B7%7D%20%5Cend%7Bgathered%7D)
The value of x is 180/7