Step 1: Recall properties of parallelograms:

• Opposite angles are congruent
• Consecutive angles are supplementary (add to 180°)

Step 2: Let's say angle A = 65°

Step 3: Find the opposite angle: Angle C = Angle A = 65° (opposite angles are equal)

Step 4: Find a consecutive angle: Consecutive angles are supplementary Angle B + Angle A = 180° Angle B + 65° = 180° Angle B = 115°

Step 5: Find the remaining angle: Angle D = Angle B = 115° (opposite angles are equal)

Step 6: Verify all angles sum to 360°: 65° + 115° + 65° + 115° = 360° ✓

Answer: The four angles are 65°, 115°, 65°, and 115°

In a parallelogram, opposite angles are congruent.

Since $\angle A$ and $\angle C$ are opposite: $\angle C = \angle A = 65°$

Answer: $\angle C = 65°$

Step 1: Recall rectangle properties:

• Opposite sides are equal
• All angles are 90°

Step 2: Use the perimeter formula: Perimeter = 2(length + width)

Step 3: Substitute known values: 40 = 2(length + 8)

Step 4: Solve for length: 40 = 2·length + 16 24 = 2·length length = 12

Step 5: Verify: Perimeter = 2(12 + 8) = 2(20) = 40 ✓

Answer: The length is 12 cm

The midsegment of a trapezoid equals the average of the bases:

$M = \frac{b_1 + b_2}{2}$

$M = \frac{8 + 14}{2} = \frac{22}{2} = 11$

Answer: The midsegment is 11

Step 1: Recall rhombus diagonal properties:

• Diagonals bisect each other at right angles
• Diagonals create four congruent right triangles

Step 2: Find half of each diagonal: Half of first diagonal = 16/2 = 8 cm Half of second diagonal = 12/2 = 6 cm

Step 3: Use Pythagorean Theorem: Each half-diagonal forms a leg of a right triangle The side of the rhombus is the hypotenuse

8² + 6² = side² 64 + 36 = side² 100 = side²

Step 4: Solve for the side: side = √100 side = 10 cm

Step 5: Verify using Pythagorean triple: This is a 6-8-10 triangle ✓

Answer: Each side of the rhombus is 10 cm

Step 1: Identify the trapezoid type: This is an isosceles trapezoid (equal legs)

Step 2: Recall isosceles trapezoid properties:

• Base angles are congruent
• The angles on each leg are supplementary

Step 3: Identify the given angle: One base angle = 70°

Step 4: Find the other angle on the same base: In an isosceles trapezoid, base angles are equal Other angle on that base = 70°

Step 5: Find angles on the other base: Angles on a leg are supplementary 70° + angle = 180° angle = 110°

Both angles on the other base = 110°

Step 6: Verify angle sum: 70° + 70° + 110° + 110° = 360° ✓

Answer: The four angles are 70°, 70°, 110°, and 110°

In a rhombus, diagonals are perpendicular and bisect each other.

The diagonals split the rhombus into 4 right triangles.

Half-diagonals:

• Half of PR = $16/2 = 8$
• Half of QS = $12/2 = 6$

Use Pythagorean Theorem: $s^2 = 8^2 + 6^2$ $s^2 = 64 + 36 = 100$ $s = 10$

Answer: Each side is 10

Step 1: List the given properties:

• AB ∥ CD (one pair of opposite sides parallel)
• AB = CD (those parallel sides are equal)
• AD = BC (other pair of opposite sides are equal)
• All angles = 90°

Step 2: Check for parallelogram:

• Opposite sides are parallel (AB ∥ CD, and AD ∥ BC implied)
• Opposite sides are equal YES, it's a parallelogram ✓

Step 3: Check for rectangle:

• It's a parallelogram with all right angles YES, it's a rectangle ✓

Step 4: Check for rhombus:

• Need all four sides equal We have AB = CD and AD = BC But we don't know if AB = AD Not necessarily a rhombus

Step 5: Check for square:

• Would need to be both rectangle AND rhombus
• We confirmed rectangle
• Need all sides equal for square Given info doesn't guarantee AB = AD Not necessarily a square

Step 6: Most specific classification: RECTANGLE is the most specific classification we can make

If we were also told AB = AD (or all sides equal), then it would be a SQUARE

Answer: This quadrilateral is a RECTANGLE. It's a parallelogram with all right angles, but we don't have enough information to prove all sides are equal (which would make it a square).