Q1. Given that Angle OUR is congruent to Angle PUN and Angle OUR =2x+5,Angle RUN=4x-2 and Angle PUT= x+32,find the measure of Angle PUN
Q2. Given that Angle TUN and Angle PUN are complementary angles and Abgle OUR = 4<RUP,what is the measure of Angle OUR.
To find the measure of Angle PUN, we can set up an equation:
Angle OUR = Angle PUN
Since Angle OUR = 2x + 5, we can substitute this into the equation:
2x + 5 = Angle PUN
Now let's solve for x.
To do this, we need another equation involving Angle PUN. We are given that Angle RUN = 4x - 2 and Angle PUT = x + 32. Since Angle PUN is congruent to Angle OUR, and we know the measurements of Angle PUT and Angle PUN, we can set up an equation:
Angle PUT = Angle PUN
Substituting the values:
x + 32 = 2x + 5
Now we can solve for x. Subtracting x from both sides gives us:
32 = x + 5
Subtracting 5 from both sides gives us:
27 = x
Now that we have the value of x, we can substitute it into the equation we set up earlier:
2x + 5 = Angle PUN
Substituting x = 27, we get:
2(27) + 5 = Angle PUN
54 + 5 = Angle PUN
59 = Angle PUN
Therefore, the measure of Angle PUN is 59 degrees.
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To find the measure of Angle OUR, we are given that Angle OUR = 4 times the measure of <RUP.
So, if we let the measure of <RUP be y, then the measure of Angle OUR would be 4y.
But we are also given that Angle TUN and Angle PUN are complementary angles. Complementary angles add up to 90 degrees. Since Angle TUN and Angle PUN are complementary, we have:
Angle TUN + Angle PUN = 90
And since Angle PUN = 4y, we can substitute this into the equation:
Angle TUN + 4y = 90
Now we can solve for y.
We can also use the fact that Angle OUR = 4y to write:
4y = Angle OUR
Substituting this into the equation we set up earlier:
Angle TUN + Angle OUR = 90
Angle TUN + 4y = 90
Now we have two equations:
Angle TUN + 4y = 90
4y = Angle OUR
Substituting the value of Angle OUR, we get:
Angle TUN + 4y = 90
Now we can solve for y. Subtracting 4y from both sides gives us:
Angle TUN = 90 - 4y
Now we can substitute this into the equation:
Angle TUN + 4y = 90
(90-4y) + 4y = 90
Simplifying the equation gives us:
90 - 4y + 4y = 90
Combining like terms gives us:
90 = 90
This equation is true, which means that y can be any value. So, we cannot determine the exact measure of Angle OUR with the given information.
To find the measure of Angle PUN in Q1, we need to set up an equation using the given information.
Here's how we can do it:
Step 1: Recall that if two angles are congruent, they have the same measure. So, we can set up the equation: Angle OUR = Angle PUN.
Step 2: Use the given measurements: Angle OUR = 2x + 5.
Step 3: Substitute the value of Angle OUR into the equation: 2x + 5 = Angle PUN.
Therefore, the measure of Angle PUN is 2x + 5.
To find the measure of Angle OUR in Q2, we'll need to use the fact that Angle OUR = 4<RUP.
Here's how we can solve it:
Step 1: We know that Angle OUR = 4<RUP.
Step 2: Use the given information: Angle OUR = 4<RUP.
Step 3: We can conclude that Angle OUR = Angle RUP because the measure of an angle and its reference angle (RUP) are equal.
Hence, the measure of Angle OUR is 4 times the measure of Angle RUP.