Question

15 fill in the chart below
property
parallelogram
rectangle
rhombus
square
isosceles trapezoid
kite
side relationships
opposite sides are parallel
only one pair of opposite sides is parallel
opposite sides are congruent
only one pair of opposite sides is congruent
all sides are congruent
two pairs of consecutive sides are congruent
angle relationships
opposite angles are congruent
at least one pair of opposite angles is congruent
all angles are right angles
diagonal relationships
diagonals bisect each other
diagonals are perpendicular
diagonals are congruent
16 if diameter ( overline{fh} ) intersects ( overline{eg} ) at a right angle, how does the length of ( overline{ei} ) compare to the length of ( overline{ig} )?
17 if ( overline{kp} cong overline{ln} ), how does the length of ( overline{oq} ) compare to the length of ( overline{ro} )?

Answer:

Question 15: Fill in the Chart (Quadrilateral Properties)

To fill the chart, we use the definitions and properties of each quadrilateral:

1. Parallelogram

• Opposite sides are parallel: ✔️
• Only one pair of opposite sides parallel: ❌
• Opposite sides are congruent: ✔️
• Only one pair of opposite sides congruent: ❌
• All sides congruent: ❌ (unless it’s a rhombus/ square)
• Two pairs of consecutive sides congruent: ❌ (unless a rhombus/ square)
• Opposite angles congruent: ✔️
• At least one pair of opposite angles congruent: ✔️ (since opposite angles are congruent)
• All angles right angles: ❌ (unless a rectangle/ square)
• Diagonals bisect each other: ✔️
• Diagonals perpendicular: ❌ (unless a rhombus/ square)
• Diagonals congruent: ❌ (unless a rectangle/ square)

2. Rectangle

• Opposite sides parallel: ✔️ (it’s a parallelogram)
• Only one pair parallel: ❌
• Opposite sides congruent: ✔️ (parallelogram property)
• Only one pair congruent: ❌
• All sides congruent: ❌ (unless a square)
• Two pairs consecutive sides congruent: ❌ (unless a square)
• Opposite angles congruent: ✔️ (parallelogram property)
• At least one pair opposite angles congruent: ✔️
• All angles right angles: ✔️
• Diagonals bisect each other: ✔️ (parallelogram property)
• Diagonals perpendicular: ❌ (unless a square)
• Diagonals congruent: ✔️

3. Rhombus

• Opposite sides parallel: ✔️ (parallelogram)
• Only one pair parallel: ❌
• Opposite sides congruent: ✔️ (parallelogram)
• Only one pair congruent: ❌
• All sides congruent: ✔️
• Two pairs consecutive sides congruent: ✔️ (all sides congruent)
• Opposite angles congruent: ✔️ (parallelogram)
• At least one pair opposite angles congruent: ✔️
• All angles right angles: ❌ (unless a square)
• Diagonals bisect each other: ✔️ (parallelogram)
• Diagonals perpendicular: ✔️
• Diagonals congruent: ❌ (unless a square)

4. Square

• Opposite sides parallel: ✔️ (parallelogram, rectangle, rhombus)
• Only one pair parallel: ❌
• Opposite sides congruent: ✔️ (parallelogram, rectangle)
• Only one pair congruent: ❌
• All sides congruent: ✔️ (rhombus)
• Two pairs consecutive sides congruent: ✔️ (all sides congruent)
• Opposite angles congruent: ✔️ (parallelogram, rectangle)
• At least one pair opposite angles congruent: ✔️
• All angles right angles: ✔️ (rectangle)
• Diagonals bisect each other: ✔️ (parallelogram)
• Diagonals perpendicular: ✔️ (rhombus)
• Diagonals congruent: ✔️ (rectangle)

5. Isosceles Trapezoid

• Opposite sides parallel: ❌ (only one pair: the two bases)
• Only one pair of opposite sides parallel: ✔️
• Opposite sides congruent: ❌ (legs are congruent, bases are parallel but not necessarily congruent)
• Only one pair of opposite sides congruent: ✔️ (the legs)
• All sides congruent: ❌
• Two pairs of consecutive sides congruent: ❌
• Opposite angles congruent: ❌ (base angles are congruent)
• At least one pair of opposite angles congruent: ✔️ (base angles; also, consecutive angles between bases are supplementary)
• All angles right angles: ❌ (unless a rectangle, but it’s a trapezoid)
• Diagonals bisect each other: ❌
• Diagonals perpendicular: ❌ (unless a special case)
• Diagonals congruent: ✔️

6. Kite

• Opposite sides parallel: ❌
• Only one pair of opposite sides parallel: ❌
• Opposite sides congruent: ❌
• Only one pair of opposite sides congruent: ❌ (two distinct pairs of consecutive sides congruent)
• All sides congruent: ❌ (unless a rhombus)
• Two pairs of consecutive sides congruent: ✔️
• Opposite angles congruent: ❌ (one pair of opposite angles congruent)
• At least one pair of opposite angles congruent: ✔️ (the angles between the unequal sides)
• All angles right angles: ❌
• Diagonals bisect each other: ❌
• Diagonals perpendicular: ✔️
• Diagonals congruent: ❌

Question 16: Diameter \( \overline{FH} \) intersects \( \overline{EG} \) at right angles

By the Perpendicular Chord Bisector Theorem, if a diameter (or radius) is perpendicular to a chord, it bisects the chord. Thus, \( \overline{FH} \perp \overline{EG} \) implies \( EI = IG \).

Question 17: If \( \overline{KP} \cong \overline{LN} \), compare \( OQ \) and \( OR \)

In a circle, if two chords are congruent, their distances from the center are equal. Since \( \overline{KP} \cong \overline{LN} \), the perpendicular distances from the center \( O \) to these chords (\( OQ \) and \( OR \)) are equal. Thus, \( OQ = OR \).

Final Answers (Chart Summary - Key Checks):

Property	Parallelogram	Rectangle	Rhombus	Square	Isosceles Trapezoid	Kite
Only one pair parallel	❌	❌	❌	❌	✔️	❌
Opposite sides congruent	✔️	✔️	✔️	✔️	❌	❌
All sides congruent	❌	❌	✔️	✔️	❌	❌
Two pairs consecutive sides congruent	❌	❌	✔️	✔️	❌	✔️
Opposite angles congruent	✔️	✔️	✔️	✔️	❌	❌
All angles right angles	❌	✔️	❌	✔️	❌	❌
Diagonals bisect each other	✔️	✔️	✔️	✔️	❌	❌
Diagonals perpendicular	❌	❌	✔️	✔️	❌	✔️
Diagonals congruent	❌	✔️	❌	✔️	✔️	❌

1. \( EI = IG \) (they are equal in length).
2. \( OQ = OR \) (they are equal in length).