Let's call the angles formed by the two intersecting lines $a$, $b$, $c$, and $d$.
Since two lines intersect, they form four angles: two pairs of opposite angles.
Since we know that three out of the four angles add up to $270^\circ$, we can set up an equation:
$a + b + c = 270^\circ$
Since $a$, $b$, $c$, and $d$ are all angles formed by the intersection of two lines, we know that $a + b + c + d = 360^\circ$.
We can now solve for $d$:
$a + b + c + d = 360^\circ$
$270^\circ + d = 360^\circ$
$d = 360^\circ - 270^\circ$
$d = 90^\circ$
So, one of the four angles formed by the intersection of the two lines is $90^\circ$.
Since opposite angles are congruent:
$a = d = 90^\circ$
$b = c = 90^\circ$
Therefore, the measures of the angles formed by the two intersecting lines are $\boxed{90^\circ, 90^\circ, 90^\circ, 90^\circ}$.
Find the measures of the angles formed by two intersecting lines if the sum of the measures of three out of the four angles is 270˚.
3 answers
NOOO!
I apologize for the incorrect response. Let's re-approach the problem.
Let's call the angles formed by the two intersecting lines $a$, $b$, $c$, and $d$.
Since two lines intersect, they form four angles: two pairs of opposite angles. Let's label the angles as shown below:
\[\angle a\qquad\angle b\]
\[\angle c\qquad\angle d\]
We are given that the sum of the measures of three out of the four angles is $270^\circ$. We can set up an equation:
\[\angle a + \angle b + \angle c = 270^\circ\]
Since opposite angles are congruent, we have:
\[\angle a = \angle d\]
\[\angle b = \angle c\]
We can now express the sum of the measures of all four angles in terms of $\angle a$:
$360^\circ = \angle a + \angle b + \angle c + \angle d$
$360^\circ = \angle a + \angle b + \angle b + \angle a$
$360^\circ = 2(\angle a + \angle b)$
Dividing both sides by 2, we have:
$180^\circ = \angle a + \angle b$
Now we have a system of equations:
$\begin{cases} \angle a + \angle b + \angle c = 270^\circ \\ \angle a + \angle b = 180^\circ \end{cases}$
We can solve this system of equations by substitution.
Substituting $\angle a + \angle b = 180^\circ$ into the first equation:
$(180^\circ) + \angle c = 270^\circ$
$\angle c = 90^\circ$
Since opposite angles are congruent, we have:
$\angle a = \angle d = 180^\circ - \angle b$
Putting it all together, the measures of the angles formed by the two intersecting lines are:
$\boxed{90^\circ, 90^\circ, 90^\circ, 90^\circ}$
Let's call the angles formed by the two intersecting lines $a$, $b$, $c$, and $d$.
Since two lines intersect, they form four angles: two pairs of opposite angles. Let's label the angles as shown below:
\[\angle a\qquad\angle b\]
\[\angle c\qquad\angle d\]
We are given that the sum of the measures of three out of the four angles is $270^\circ$. We can set up an equation:
\[\angle a + \angle b + \angle c = 270^\circ\]
Since opposite angles are congruent, we have:
\[\angle a = \angle d\]
\[\angle b = \angle c\]
We can now express the sum of the measures of all four angles in terms of $\angle a$:
$360^\circ = \angle a + \angle b + \angle c + \angle d$
$360^\circ = \angle a + \angle b + \angle b + \angle a$
$360^\circ = 2(\angle a + \angle b)$
Dividing both sides by 2, we have:
$180^\circ = \angle a + \angle b$
Now we have a system of equations:
$\begin{cases} \angle a + \angle b + \angle c = 270^\circ \\ \angle a + \angle b = 180^\circ \end{cases}$
We can solve this system of equations by substitution.
Substituting $\angle a + \angle b = 180^\circ$ into the first equation:
$(180^\circ) + \angle c = 270^\circ$
$\angle c = 90^\circ$
Since opposite angles are congruent, we have:
$\angle a = \angle d = 180^\circ - \angle b$
Putting it all together, the measures of the angles formed by the two intersecting lines are:
$\boxed{90^\circ, 90^\circ, 90^\circ, 90^\circ}$
1 answer