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.