Question

pythagorean theorem
(a^2 + b^2 = c^2)
((\text{leg})^2 + (\text{leg})^2 = (\text{hypotenuse})^2)
triangle 1: leg 12, leg 9, hypotenuse (x)
triangle 2: leg 12, leg (y), hypotenuse 13
properties of quadrilaterals
square, rectangle, parallelogram, rhombus
all sides are congruent
opposite sides are congruent
opposite angles are parallel (likely typo, intended opposite sides are parallel)
opposite angles are congruent

Explanation:

First Triangle (finding \( x \)):

Step1: Identify legs and hypotenuse

Legs are 12 and 9, hypotenuse is \( x \). Use Pythagorean theorem \( a^2 + b^2 = c^2 \).
\( 12^2 + 9^2 = x^2 \)

Step2: Calculate squares

\( 144 + 81 = x^2 \)

Step3: Sum and take square root

\( 225 = x^2 \), so \( x = \sqrt{225} = 15 \)

Second Triangle (finding \( y \)):

Step1: Identify legs and hypotenuse

Legs are 12 and \( y \), hypotenuse is 13. Use Pythagorean theorem \( a^2 + b^2 = c^2 \).
\( 12^2 + y^2 = 13^2 \)

Step2: Calculate squares and rearrange

\( 144 + y^2 = 169 \), so \( y^2 = 169 - 144 = 25 \)

Step3: Take square root

\( y = \sqrt{25} = 5 \)

Properties of Quadrilaterals (Filling the table):

"All sides are congruent":

• Square: Yes (all sides equal)
• Rectangle: No (opposite sides equal, not all)
• Parallelogram: No (opposite sides equal, not all)
• Rhombus: Yes (all sides equal)

"Opposite sides are congruent":

• Square: Yes (opposite sides equal)
• Rectangle: Yes (opposite sides equal)
• Parallelogram: Yes (definition of parallelogram)
• Rhombus: Yes (opposite sides equal as it's a parallelogram)

"Opposite angles are parallel" (likely a typo, should be "Opposite sides are parallel"):

• Square: Yes (opposite sides parallel)
• Rectangle: Yes (opposite sides parallel)
• Parallelogram: Yes (definition)
• Rhombus: Yes (opposite sides parallel as it's a parallelogram)

"Opposite angles are congruent":

• Square: Yes (all angles 90°, so opposite congruent)
• Rectangle: Yes (all angles 90°, opposite congruent)
• Parallelogram: Yes (opposite angles congruent)
• Rhombus: Yes (opposite angles congruent as it's a parallelogram)

Answer:

s:

• For \( x \): \( \boldsymbol{15} \)
• For \( y \): \( \boldsymbol{5} \)
• Quadrilateral Properties Table (filled):

Property	Square	Rectangle	Parallelogram	Rhombus
Opposite sides are congruent	Yes	Yes	Yes	Yes
Opposite sides are parallel	Yes	Yes	Yes	Yes
Opposite angles are congruent	Yes	Yes	Yes	Yes