Answer:
Step-by-step explanation:
a)
To solve this problem, we have to remember about the Inscribed Angle Theorem.
The Inscribed Angle Theorem states that an angle formed by two chords in a circle (an inscribed angle) is half the measure of the intercepted arc.
In this problem:
- the angle P is formed by two chords (PR and PQ), so its measure is half the measure of the arc QR.
- the angle R is formed by two chords (RP and RQ), so its measure is half the measure of the arc PQ.
- the angle Q is formed by two chords (QR and QP), so its measure is half the measure of the arc PR.
So,
m∠P = 5x/2
m∠Q = (6x + 7)/2
m∠R = (8x - 8)/2
As we know that the sum of the interior angle measures of a triangle always adds up to 180°:
m∠P + m∠Q + m∠R = 180°
5x/2 + (6x + 7)/2 + (8x - 8)/2 = 180°
( 5x + 6x + 7 + 8x - 8)/2 = 180°
(19x -1)/2 = 180°
19x -1 = 180° × 2
19x = 360° + 1°
19x = 361°
x = 361°/19
x= 19°
b)
m∠P = 5x/2 = 5 × 19 /2 = 95 /2 = 47.5°
m∠Q = (6x + 7)/2 = (6 × 19 + 7)/2 = (114+ 7)/2 = 121/2 =60.5°
m∠R = (8x - 8)/2 = (8 × 19 - 8)/2 = (152 -8)/2 = 144/2 = 72°
As all three interior angles are different, the triangle is scalene.
