SX¯¯¯¯¯¯¯¯ bisects <RST. m<RST = 10x. m<XST = 3x+16. Find m<RSX
Vertically opposite angles are equal. Therefore, we can set up the equation:
9x + 3 = 12x - 6
To solve for x, we'll start by subtracting 9x from both sides:
3 = 3x - 6
Next, we'll add 6 to both sides:
9 = 3x
Finally, we'll divide both sides by 3:
3 = x
Therefore, the value of x is 3.
A counterexample for the given conjecture would be a number that is divisible by 3 but not by 6.
So, the counterexample for the conjecture "Any number that is divisible by 3 is also divisible by 6" would be the number 9.
9 is divisible by 3, as \(9/3 = 3\), but it is not divisible by 6 since \(9/6 = 1.5\) which is not a whole number.
Since SX bisects <RST, that means the angle <RST is divided into two congruent angles. Let's call the measure of each congruent angle y.
So we have:
<RST = y
<RSX = y
<XST = 3x + 16
Since the sum of the angles in a triangle is 180 degrees, we can write the equation:
<RST + <RSX + <XST = 180
Substituting the given values:
y + y + (3x + 16) = 180
Combining like terms:
2y + 3x + 16 = 180
Subtracting 16 from both sides:
2y + 3x = 164
Since <RST = y and its measure is 10x, we can write:
y = 10x
Substituting this into the previous equation:
2(10x) + 3x = 164
Simplifying:
20x + 3x = 164
Combining like terms:
23x = 164
Dividing both sides by 23:
x = 164/23
Simplifying:
x = 7.13
Now, let's find the measure of <RSX. We substitute this value of x back into the equation for <RSX:
<RST = 10x = 10(7.13) = 71.3 degrees
Therefore, m<RSX = 71.3 degrees.
The measure of two verticle angles are 9x + 3 and 12x - 6. Find x.(1 point) Responses
3
8.7
12
9.5
What is a counterexample for the conjecture? Conjecture: Any number that is divisible by 3 is also divisible by 6. (1 point) Responses
12
26
9
30
To find the measure of angle RSX, we need to set up an equation using the given information about angles RST and XST.
First, let's write down the information we have:
m<RST = 10x
m<XST = 3x + 16
We know that the sum of angles RST and XST is 180 degrees since they form a straight line. So we can write an equation as follows:
m<RST + m<XST = 180
Substituting the given angle measures:
10x + (3x + 16) = 180
Simplifying the equation:
10x + 3x + 16 = 180
13x + 16 = 180
13x = 180 - 16
13x = 164
Dividing both sides by 13:
x = 164/13
x = 12.62
Now that we have found the value of x, we can substitute it into the expression for m<RST to find its measure:
m<RST = 10(12.62) = 126.2 degrees
To find the measure of angle RSX, we can subtract the measure of angle XST from the measure of angle RST:
m<RSX = m<RST - m<XST
m<RSX = 126.2 - (3(12.62) + 16)
m<RSX = 126.2 - (37.86 + 16)
m<RSX = 126.2 - 53.86
m<RSX = 72.34
Therefore, m<RSX = 72.34 degrees.
To find the measure of angle RSX, we need to use the angle bisector theorem. According to this theorem, when a line bisects an angle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle.
In this case, we have an angle bisector SX¯¯¯¯¯¯¯¯ that divides <RST into two smaller angles. Let's call the measure of angle RSX as y.
According to the angle bisector theorem, we can set up the following ratio:
RS / ST = RX / XT
Since SX¯¯¯¯¯¯¯¯ bisects <RST, we can set up the following proportion:
RS / ST = RX / XT
To determine the lengths of the segments RS and ST in terms of x, we need to use the given information about the measures of the angles:
m<RST = 10x
m<XST = 3x + 16
Since the angle measures of a triangle add up to 180 degrees, we have:
m<RST + m<XST + m<RSX = 180
Substituting the given angle measures:
10x + (3x + 16) + y = 180
Simplifying the equation:
13x + 16 + y = 180
To find the measure of angle RSX (y), we need to solve this equation for y.
y = 180 - 13x - 16
Simplifying further:
y = 164 - 13x
Therefore, the measure of angle RSX (m<RSX) is 164 - 13x.