In Fig. 6.16, if $ x+y=w+z $, then prove that $ \mathrm{AOB} $ is a line.
"
To do:
We have to prove that $AOB$ is a line.
Solution:
We know that,
The sum of the measures of the angles in linear pairs is always $180^o$.
So in order to prove that $AOB$ is a straight line, we have to prove that $x+y$ is a linear pair of $AOB$.
This implies,
$x+y=180^o$
We also know that,
The angles around a point give $360^o$.
This implies,
$x+y+w+z=360^o$
Since we have,
$x+y=w+z$
We get,
$2x+y=360^o$
This implies,
$x+y=\frac{360^o}{2}$
$x+y=180^o$
Therefore, $x+y$ is the linear pair
This implies, that $AOB$ is a line.
- Related Articles
- In Fig. 5.10, if \( \mathrm{AC}=\mathrm{BD} \), then prove that \( \mathrm{AB}=\mathrm{CD} \)."\n
- If \( x=a^{m+n}, y=a^{n+1} \) and \( z=a^{l+m} \), prove that \( x^{m} y^{n} z^{l}=x^{n} y^{l} z^{m} \)
- In Fig. 6.29, if \( \mathrm{AB}\|\mathrm{CD}, \mathrm{CD}\| \mathrm{EF} \) and \( y: z=3: 7 \), find \( x \)."\n
- In Fig. 6.28, find the values of \( x \) and \( y \) and then show that \( \mathrm{AB}=\mathrm{CD} \)."\n
- If \( a^{x}=b^{y}=c^{z} \) and \( b^{2}=a c \), then show that \( y=\frac{2 z x}{z+x} \).
- In Fig. 6.15, \( \angle \mathrm{PQR}=\angle \mathrm{PRQ} \), then prove that \( \angle \mathrm{PQS}=\angle \mathrm{PRT} \)"\n
- If \( x+y+z=0 \), show that \( x^{3}+y^{3}+z^{3}=3 x y z \).
- If $x^2 + y^2 = 27xy$ then prove that $2log(x-y) = 2log5 + logx + logy$.
- If \( 3^{x}=5^{y}=(75)^{z} \), show that \( z=\frac{x y}{2 x+y} \).
- If $R (x, y)$ is a point on the line segment joining the points $P (a, b)$ and $Q (b, a)$, then prove that $x + y = a + b$.
- If $R\ ( x,\ y)$ is a point on the line segment joining the points $P\ ( a,\ b)$ and $Q\ ( b,\ a)$, then prove that $a+b=x+y$
- Find x, y, z that satisfy 2/n = 1/x + 1/y + 1/z in C++
- If $x+y+z = 0$ show that $x^{3}+y^{3}+z^{3}=3xyz$
- In \( \Delta X Y Z, X Y=X Z \). A straight line cuts \( X Z \) at \( P, Y Z \) at \( Q \) and \( X Y \) produced at \( R \). If \( Y Q=Y R \) and \( Q P=Q Z \), find the angles of \( \Delta X Y Z \).
- If the continued fraction form of \( \frac{97}{19} \) is \( w+\frac{1}{x+\frac{1}{y}} \) where \( w, x, y \) are integers, then find the value of \( w+x+y \).
Kickstart Your Career
Get certified by completing the course
Get Started