4.2: The Quadrilateral Family | Geometry Honors

How it works

A parallelogram has two pairs of parallel opposite sides. Rectangles add right angles, rhombi add four equal sides, and squares satisfy both conditions. Definitions are inclusive: every square is a rectangle and a rhombus.

\text{square}\subset\text{rectangle}\subset\text{parallelogram}

Worked examples

Example 1

A parallelogram has an angle of $\displaystyle 70^\circ$ . Find the other three angles.

The opposite angle is $\displaystyle 70^\circ$ ; both adjacent angles are $\displaystyle 110^\circ$ .

Example 2

A rectangle has diagonals of lengths $\displaystyle 2x+1$ and $\displaystyle 9$ . Find $\displaystyle x$ .

Rectangle diagonals are congruent, so $\displaystyle 2x+1=9$ and $\displaystyle x=4$ .

Problems

Problem 4.2.1

A parallelogram has adjacent side lengths $\displaystyle 5$ and $\displaystyle 3$ . Find its perimeter.

A parallelogram with adjacent sides 5 and 3. — Opposite sides are parallel and congruent.

Show worked solution

$\displaystyle P=2(5+3)=16$ units. Opposite sides match.

A parallelogram with adjacent sides 5 and 3. The solution adds labeled relationships. — Each opposite side matches, so perimeter is 2(5 + 3) = 16.

Problem 4.2.2

Does a quadrilateral with perpendicular diagonals have to be a square?

Show worked solution

No. A rhombus with unequal diagonal lengths has perpendicular diagonals but is not a square.

Problem 4.2.3

A rhombus has diagonals $\displaystyle 6$ and $\displaystyle 8$ . Find its side length and area.

Show worked solution

The diagonals bisect at right angles. Each side is $\displaystyle \sqrt{3^2+4^2}=5$ . The area is $\displaystyle 6\cdot8/2=24$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.2.1

Problem 4.2.2

Problem 4.2.3

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