Question

Math help plss !! uwu im stuck.

Answer 1

Step-by-step explanation:

1) The four points are:

(x₁, y₁) = (-2, -1)

(x₂, y₂) = (3, 13)

(x₃, y₃) = (15, 5)

(x₄, y₄) = (13, -11)

Using the distanced formula the four side lengths are:

d₁₂ = √((13−-1)² + (3−-2)²) = √221

d₂₃ = √((5−13)² + (15−3)²) = √208

d₃₄ = √((-11−5)² + (13−15)²) = √260

d₄₁ = √((-1−-11)² + (-2−13)²) = √325

None of the lengths are equal, so we know this isn't a rhombus, parallelogram, or kite.  Is it a trapezoid?  To find out, let's find the slopes between the two lines that look like they might be parallel.

m₂₃ = (5 - 13) / (15 - 3) = -2/3

m₄₁ = (-1−-11) / (-2−13) = -2/3

They are indeed parallel.  So this is a trapezoid.

2) Given:

PS ≅ QR

m∠P + m∠Q = 180

m∠R + m∠S = 180

∠P ≅ ∠S

By converse of Alternate Interior Angles Theorem, since ∠P and ∠Q are supplementary, line PS and QR must be parallel.

If a quadrilateral has one pair of opposite sides that are both parallel and congruent, then it is a parallelogram.

Adjacent angles of a parallelogram are supplementary, so m∠P + m∠S = 180.

Since ∠P ≅ ∠S, then by definition of congruent angles, m∠P = m∠S.

Substitution:

m∠P + m∠P = 180

m∠P = 90

Substitution:

m∠S = 90

Opposite angles of a parallelogram are congruent, so m∠Q = m∠S = 90 and m∠R = m∠P = 90.

A parallelogram with four right angles is a rectangle.