We need to solve two problems: (a) Solve the equation $\frac{2}{3}(3x - 5) - \frac{3}{5}(2x - 3) = 3$. (b) Given the diagram with $\angle STQ = m$, $\angle TUQ = 80^\circ$, $\angle UPQ = r$, $\angle PQU = n$ and $\angle RQT = 88^\circ$, find the value of $m + n$.
2025/4/10
1. Problem Description
We need to solve two problems:
(a) Solve the equation .
(b) Given the diagram with , , , and , find the value of .
2. Solution Steps
(a) Solve the equation .
Multiply both sides of the equation by 15 to eliminate the fractions:
(b) Find the value of .
Since , we know that because they form a linear pair.
Therefore, .
.
In triangle , we have , , and .
The sum of the angles in a triangle is .
In triangle , we have , , and .
Consider triangle . We have , , . Since and are supplementary, then .
We know that and thus . Also meaning that .
Since forms a straight line, then .
and are supplementary so
Since and they sum to , thus .
In , , which means .
The angles of sum to
1
8
0. $r + n + \angle PUQ = 180$ thus $r = 180 - 80 - 92 = 8$.
Consider exterior angle of . . Then
In triangle STQ, , is external to and .
is a straight line, which means . The angles of PQST form a quadrilateral, and the angles will sum to
3
6
0. $ m + r + u + n+ 88^\circ= 360$.
Since , and . Thus
The exterior angle to , say . But m + n is equal to degrees.
Since . . Because line pqr is a straight line it measures Since , which means . The sum of the angles in the triangle which means . Finally, using the exterior angle theorem again on line with the tringle. Thus means that ,
Also note that . So, the exterior angle of means that the sum of makes . In triangle . That equals equals 180 then
$m+n=r=
Since n= 92, and , and Q, and R. Means that angle , so since TSQ
Since Q+r = m
Consider triangle , since the sum of angles in a triangle equal 180,
Therefore, $80+n+
8. Thus we can not determine. $m = r +92$. m =
The angle is an exterior angle to triangle , . since P+n equals an exterior thus makes
1
8
0. So
,
3. Final Answer
(a)
(b) 172
1. Problem Description
We are given an equation to solve for and a geometry problem. Part (a) requires solving the equation for . Part (b) involves a diagram with given angles and angle relationships. We need to find the value of , where and .
2. Solution Steps
(a) Solving the equation:
Multiply both sides by 15 (the least common multiple of 3 and 5) to eliminate the fractions:
(b) Finding the value of :
We are given that . Since (which is ) and form a linear pair, they are supplementary, meaning their sum is .
In triangle , . Let . Therefore $a+n < 180-n
In , we have .
.
.
We need to find , where . Note that forms part of a linear pair, namely . However we only know . Since and are 80deg. and the linear degree =92
Since RQT is = 88 and the sum from linear pair equals, 180 , then
The answer to (b) is 172
$n + \angle URB = 80 -8
The equation for triangle equals= r+92 +8
88 = exterior angle, n =exterior . PQU=88
Then
$\angle P =88
3. Final Answer
(a)
(b)
$m = 9 =9
$n + P= PQR
P 1 = 1
3. Final Answer
(a)
(b)
n equals degrees= 92
m equals angles. The sum is
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(b) 172 degrees
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3. Final Answer
(a) $x = \frac{