Question

1. find the values of $x$, $y$, and $z$ given the triangle diagram with angles $43^\circ$, $24^\circ$, $57^\circ$

Explanation:

Step1: Solve for $z$ (triangle angle sum)

The sum of angles in a triangle is $180^\circ$. For the right triangle:
$z + 57^\circ + (43^\circ + 24^\circ) = 180^\circ$
$z + 57^\circ + 67^\circ = 180^\circ$
$z = 180^\circ - 124^\circ = 56^\circ$

Step2: Solve for $y$ (supplementary to $z$)

$y$ and $z$ are supplementary (linear pair):
$y + z = 180^\circ$
$y = 180^\circ - 56^\circ = 124^\circ$

Step3: Solve for $x$ (triangle angle sum)

For the left triangle:
$x + y + 43^\circ = 180^\circ$
$x + 124^\circ + 43^\circ = 180^\circ$
$x = 180^\circ - 167^\circ = 13^\circ$

Answer:

$x = 13^\circ$, $y = 124^\circ$, $z = 56^\circ$