SAT Properties of Triangles

Correct Answer: C

Solution 1:

Tip: The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.
$\angle BAC$ and $\angle x$ are supplementary angles. Therefore, their measures add to $180^\circ$ and

$m\angle BAC = 180-x.$

$\angle y$ is exterior, and therefore its measure equals the sum of the measures of the two remote interior angles:

$\begin{aligned} y &= m\angle BAC + z\\ y &= 180-x + z\\ x+y-180 &= z\\ \end{aligned}$

Solution 2:

Tip: The measures of the angles in a triangle add to $180^\circ.$
$\angle BAC$ and $\angle x$ are supplementary angles. $\angle BCA$ and $\angle y$ are supplementary angles too. From the definition of supplementary angles, it follows that

$m\angle BAC = 180-x$ and
$m\angle BCA = 180-y$

The measures of the angles in a triangle add to $180^\circ.$ Therefore,

$\begin{array}{r c l} m\angle ABC + m\angle BAC + m\angle BCA &=& 180\\ m\angle ABC &=& 180- m\angle BAC - m\angle BCA\\ z &=& 180 - (180-x) - (180-y)\\ z &=& 180- 180 + x - 180 + y \\ z &=& x + y - 180\\ \end{array}$

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Incorrect Choices:

(A)
If you think that $x, y,$ and $z$ add to 180 $^\circ,$ you will get this wrong answer. $x$ and $y$ are exterior, and not interior, angles.

(B)
You will get this wrong answer if in Solution 1 you add 180 to both sides instead of subtract it from both sides, like this:

$\begin{aligned} y &= 180-x + z\\ x+y \fbox{+}180 &= z \quad \text{mistake: added instead of subtracted}\\ \end{aligned}$

or if in Solution 2 you add $m\angle BCA$ to both sides instead of subtract it, like this:

$\begin{aligned} &m\angle ABC + m\angle BAC + m\angle BCA =180\\ &m\angle ABC = 180- m\angle BAC \fbox{+} m\angle BCA \quad \text{mistake: added, instead of subtracted}\\ &z = 180 - (180-x) \fbox{+} (180-y)\\ \end{aligned}$

(D)
If you apply to $\angle z$ the theorem which states that the measure of an exterior angle in a triangle equals the sum of the measures of the two nonadjacent interior angles, you will get this wrong answer. Note that $\angle z$ is an interior angle, and $\angle x$ and $\angle y$ are exterior angles.

(E)
This wrong choice is the sum of the measures of $\angle BAC$ and $\angle BCA.$ It is offered to confuse you.

x = z+180-y, or z= x+y – 180.