Question

1. find the values of x and y that make these triangles congruent by the hl theorem. x = 2, y = 1 x = 1, y = 2 x = 3, y = 2 x = 2, y = 3

Explanation:

Step1: Set up equations from HL - Theorem

For right - triangles to be congruent by the HL (Hypotenuse - Leg) Theorem, the hypotenuses are equal and the legs are equal. So we have the following two equations:
$3y=x + 3$ (equating hypotenuses) and $y + 1=x$ (equating legs).

Step2: Substitute $x=y + 1$ into the first equation

Substitute $x$ in the equation $3y=x + 3$ with $y + 1$. We get $3y=(y + 1)+3$.
Simplify the right - hand side: $3y=y+4$.
Subtract $y$ from both sides: $3y−y=y + 4−y$, which gives $2y=4$.
Divide both sides by 2: $y = 2$.

Step3: Find the value of $x$

Substitute $y = 2$ into the equation $x=y + 1$. Then $x=2 + 1=3$.

Answer:

$x = 3,y = 2$ (corresponds to the third option in the multiple - choice list)