What is the measure of ∠LPQ? What is the measure of ∠LPN

Mathematics

2 answers:

77julia77 [94]3 years ago

6 0

$\tt Step-by-step~explanation:$

Since no real values are given for the angles, we will not have an exact angle measurement.

If ∠LPQ is x+28, then that means ∠NPM is the same, since they are vertical angles.

$\tt LPQ=x+28\\NPM=x+28$

Since ∠QPM is x, then ∠LPN is also x, because they are also vertical angles.

$\tt QPM=x\\LPN=x$

$\large\boxed{\tt Our ~final~answer:~LPQ=x+28,~LPN=x}$

Alenkinab [10]3 years ago

3 0

Lpq=x ¥€ % 28^ £¥[ x 671

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Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.

mr_godi [17]

Answer:

See proof below

Step-by-step explanation:

Let $r=\frac{w+z}{1+wz}$ . If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.

Remember this useful identity: if x is a complex number then $x\bar{x}=|x|^2$ where $\bar{x}$ is the conjugate of x.

Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):

$\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)$

= $\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1$