Respuesta :
Step-by-step explanation:
Notice that, the angle QRS is external to the triangle and adjacent to the angle PRQ. According to the theorem of a external/adjacent angle, we have: m∠QRS = m∠PQR + m∠RPQ, where PQR and RPQ are internal angles.
From the hypothesis, we have:
m∠QRS =(10x−12)∘(10x−12)
m∠PQR = (3x+20)∘(3x+20)
m∠RPQ=(3x−8)∘(3x−8)
Using the first equation and replacing the hypothesis:
m∠QRS = m∠PQR + m∠RPQ
(10x−12)∘(10x−12) = (3x+20)∘(3x+20) + (3x−8)∘(3x−8)
Multiplying and applying the remarkable identity:
[tex](10x - 12)^{2} = (3x+20)^{2} + (3x - 8)^{2} \\(10x)^{2} - 2(10x)(12) + (12)^{2} = (3x)^{2} + 2(3x)(20) + (20)^{2} + (3x)^{2} - 2(3x)(8) + (8)^{2} \\100x^{2} -240x+144=9x^{2} +120x+400 + 9x^{2} -48x+64\\100x^{2} -240x+144-18x^{2} -72x-464=0\\82x^{2} -312x-320=0\\[/tex]
Then, we use a calculator to find the roots, which are:
[tex]x=4.7\\x=-0.8[/tex]
In this case, we will see what root is the right one.
Now, we replace it into m∠QRS =(10x−12)∘(10x−12), because we need to find m∠QRS.
m∠QRS =(10x−12)∘(10x−12) = (10(4.7) - 12) (10(4.7) - 12) = (35) (35) = 1225
Answer:
100.5°
Step-by-step explanation:
Data
m∠QRS = (10x−12)°
m∠PQR = (3x+20)°
m∠RPQ = (3x−8)°
The three angles form the triangle ΔPQR, then the addition of them makes 180°.
m∠QRS + m∠PQR + m∠RPQ = 180°
10x−12 + 3x+20 + 3x−8 = 180
16x = 180
x = 180/16 = 11.25
Then, m∠QRS = 10*11.25−12 = 100.5°
