Question

what is the value of x​

Answer 1

Step-by-step explanation:

Since AB = AC, and ∠BAC = 80°, we must have ∠ACB = ∠ABC = 50°. Then ∠PBC = 40°.

Construct AD such that AD = AB = AC and DAC = 40°. Let E be the intersection of AD and BC. In AEC, two angles are 40° and 50°, so AEC = 90°.

Construct DB and DC. Then ABC and ADC are congruent isosceles triangles (by side-angle-side), each having a vertex angle of 20°, and each having equal angles of 70° each. Thus ∠ADB = ∠ABD = ∠ADC = ∠ACD = 70°. Consequently ∠EBD = ∠DCE = 20°. Also we have DB = DC.

Now construct DP. Then BAP and ADP are congruent triangles (by side-angle-side), so ∠ADP = 10°, meaning ∠PDB = 60°. Since ∠PDB = ∠DBP = 60°, we have ∠BPD = 60°, and triangle PDB is an equilateral triangle. Thus PD = PB = BD, and they are all equal to DC since BD = DC.

Consider triangle DPC. It is an isosceles triangle with a vertex angle equal to 80°, so its equal angles are each 50°. Thus ∠DPC = ∠DCP = 50°, and ∠ECP = ∠DCP – ∠DCE = 50° – 20° = 30°. Then ∠ACP = ∠ACB – ∠ECP = 50° – 30° = 20°.

Finally we consider triangle APC. We have x = ∠ACP = 180° – ∠PAC – ∠ACP = 180° – 60° – 20° = 100°.

Answer 2

Step-by-step explanation:

by exterior angle property,

angle PBA+ angle BAP=X

20+10

=30°