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# 6.2: Properties of Parallelograms

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Define a parallelogram.
• Understand the properties of a parallelogram
• Apply theorems about a parallelogram’s sides, angles and diagonals.

## Review Queue

1. Draw a quadrilateral with one set of parallel sides.
2. Draw a quadrilateral with two sets of parallel sides.
3. Find the measure of the missing angles in the quadrilaterals below.

Know What? A college has a parallelogram-shaped courtyard between two buildings. The school wants to build two walkways on the diagonals of the parallelogram and a fountain where they intersect. The walkways are going to be 50 feet and 68 feet long. Where would the fountain be?

What is a Parallelogram?

Parallelogram: A quadrilateral with two pairs of parallel sides.

Notice that each pair of sides is marked parallel. Also, recall that two lines are parallel when they are perpendicular to the same line. Parallelograms have a lot of interesting properties.

Investigation 6-2: Properties of Parallelograms

Tools Needed: Paper, pencil, ruler, protractor

1. Draw a set of parallel lines by drawing a 3 inch line on either side of your ruler.
2. Rotate the ruler and repeat so you have a parallelogram. If you have colored pencils, outline the parallelogram in another color.
3. Measure the four interior angles of the parallelogram as well as the length of each side. What do you notice?
4. Draw the diagonals. Measure each and then measure the lengths from the point of intersection to each vertex.

To continue to explore the properties of a parallelogram, see the website: http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php

Opposite Sides Theorem: If a quadrilateral is a parallelogram, then the opposite sides are congruent.

If then

Opposite Angles Theorem: If a quadrilateral is a parallelogram, then the opposite angles are congruent.

If then

Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.

If then

mA+mD=180mA+mB=180mB+mC=180mC+mD=180\begin{align*}m \angle A + m \angle D = 180^\circ\\ m \angle A + m \angle B = 180^\circ\\ m \angle B + m \angle C = 180^\circ\\ m \angle C + m \angle D = 180^\circ\end{align*}

Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.

If then

Proof of Opposite Sides Theorem

Given: ABCD\begin{align*}ABCD\end{align*} is a parallelogram with diagonal BD¯¯¯¯¯¯¯¯\begin{align*}\overline{BD}\end{align*}

Prove: AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯,AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{DC},\overline{AD} \cong \overline{BC}\end{align*}

Statement Reason
1. ABCD\begin{align*}ABCD\end{align*} is a parallelogram with diagonal BD¯¯¯¯¯¯¯¯\begin{align*} \overline{BD}\end{align*} Given
2. AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯,AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \| \overline{DC}, \overline{AD} \| \overline{BC}\end{align*} Definition of a parallelogram
3. ABDBDC,ADBDBC\begin{align*}\angle ABD \cong \angle BDC, \angle ADB \cong \angle DBC\end{align*} Alternate Interior Angles Theorem
4. DB¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯\begin{align*}\overline{DB} \cong \overline{DB}\end{align*} Reflexive PoC
5. \begin{align*}\triangle ABD \cong \triangle CDB\end{align*} ASA
6. \begin{align*}\overline{AB} \cong \overline{DC}, \overline{AD} \cong \overline{BC}\end{align*} CPCTC

The proof of the Opposite Angles Theorem is almost identical. For the last step, the angles are congruent by CPCTC.

Example 1: \begin{align*}ABCD\end{align*} is a parallelogram. If \begin{align*}m \angle A = 56^\circ\end{align*}, find the measure of the other angles.

Solution: Draw a picture. When labeling the vertices, the letters are listed, in order, clockwise.

If \begin{align*}m \angle A = 56^\circ\end{align*}, then \begin{align*}m \angle C = 56^\circ\end{align*} by the Opposite Angles Theorem.

\begin{align*}m \angle A + m \angle B & = 180^\circ \quad \text{by the Consecutive Angles Theorem.}\\ 56^\circ + m \angle B & = 180^\circ\\ m \angle B & = 124^\circ \quad m \angle D = 124^\circ \quad \text{because it is opposite angles with} \ \angle B.\end{align*}

Example 2: Algebra Connection Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

Solution: Opposite sides are congruent.

\begin{align*}6x - 7 & = 2x + 9 && y + 3 = 12\\ 4x & = 16 && \qquad y = 9\\ x & = 4\end{align*}

## Diagonals in a Parallelogram

From the Parallelogram Diagonals Theorem, we know that the diagonals of a parallelogram bisect each other.

Example 3: Show that the diagonals of \begin{align*}FGHJ\end{align*} bisect each other.

Solution: Find the midpoint of each diagonal.

\begin{align*}&\text{Midpoint of} \ \overline{FH}: \qquad \left ( \frac{-4 + 6 }{2}, \frac{5 - 4}{2} \right ) = (1, 0.5)\\ &\text{Midpoint of} \ \overline{GJ}: \qquad \left ( \frac{3 - 1}{2}, \frac{3 - 2}{2} \right ) = (1,0.5)\end{align*}

Because they are the same point, the diagonals intersect at each other’s midpoint. This means they bisect each other.

This is one way to show a quadrilateral is a parallelogram.

Example 4: Algebra Connection \begin{align*}SAND\end{align*} is a parallelogram and \begin{align*}SY = 4x - 11\end{align*} and \begin{align*}YN = x + 10\end{align*}. Solve for \begin{align*}x\end{align*}.

Solution:

\begin{align*}SY & = YN\\ 4x - 11 & = x + 10\\ 3x & = 21\\ x & = 7\end{align*}

Know What? Revisited The diagonals bisect each other, so the fountain is going to be 34 feet from either endpoint on the 68 foot diagonal and 25 feet from either endpoint on the 50 foot diagonal.

## Review Questions

• Questions 1-6 are similar to Examples 2 and 4.
• Questions 7-10 are similar to Example 1.
• Questions 11-23 are similar to Examples 2 and 4.
• Questions 24-27 are similar to Example 3.
• Questions 28 and 29 are similar to the proof of the Opposite Sides Theorem.
• Question 30 is a challenge. Use the properties of parallelograms.

\begin{align*}ABCD\end{align*} is a parallelogram. Fill in the blanks below.

1. If \begin{align*}AB = 6\end{align*}, then \begin{align*}CD =\end{align*} ______.
2. If \begin{align*}AE = 4\end{align*}, then \begin{align*}AC =\end{align*} ______.
3. If \begin{align*}m \angle ADC = 80^\circ, m \angle DAB\end{align*} = ______.
4. If \begin{align*}m \angle BAC = 45^\circ, m \angle ACD\end{align*} = ______.
5. If \begin{align*}m \angle CBD = 62^\circ, m \angle ADB\end{align*} = ______.
6. If \begin{align*}DB = 16\end{align*}, then \begin{align*}DE\end{align*} = ______.
7. If \begin{align*}m \angle B = 72^\circ\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, find the other three angles.
8. If \begin{align*}m \angle S = 143^\circ\end{align*} in parallelogram \begin{align*}PQRS\end{align*}, find the other three angles.
9. If \begin{align*}\overline{AB} \perp \overline{BC}\end{align*} in parallelogram \begin{align*}ABCD\end{align*}, find the measure of all four angles.
10. If \begin{align*}m \angle F = x^\circ\end{align*} in parallelogram \begin{align*}EFGH\end{align*}, find the other three angles.

For questions 11-19, find the measures of the variable(s). All the figures below are parallelograms.

Use the parallelogram \begin{align*}WAVE\end{align*} to find:

1. \begin{align*}m \angle AWE\end{align*}
2. \begin{align*}m \angle ESV\end{align*}
3. \begin{align*}m \angle WEA\end{align*}
4. \begin{align*}m \angle AVW\end{align*}

Find the point of intersection of the diagonals to see if \begin{align*}EFGH\end{align*} is a parallelogram.

1. \begin{align*}E(-1, 3), F(3, 4), G(5, -1), H(1, -2)\end{align*}
2. \begin{align*}E(3, -2), F(7, 0), G(9, -4), H(5, -4)\end{align*}
3. \begin{align*}E(-6, 3), F(2, 5), G(6, -3), H(-4, -5)\end{align*}
4. \begin{align*}E(-2, -2), F(-4, -6), G(-6, -4), H(-4, 0)\end{align*}

Fill in the blanks in the proofs below.

1. Opposite Angles Theorem

Given: \begin{align*}ABCD\end{align*} is a parallelogram with diagonal \begin{align*}\overline{BD}\end{align*}

Prove: \begin{align*}\angle A \cong \angle C\end{align*}

Statement Reason
1. Given
2. \begin{align*}\overline{AB} \| \overline{DC},\overline{AD} \| \overline{BC}\end{align*}
3. Alternate Interior Angles Theorem
4. Reflexive PoC
5. \begin{align*}\triangle ABD \cong \triangle CDB\end{align*}
6. \begin{align*}\angle A \cong \angle C\end{align*}
1. Parallelogram Diagonals Theorem

Given: \begin{align*}ABCD\end{align*} is a parallelogram with diagonals \begin{align*}\overline{BD}\end{align*} and \begin{align*}\overline{AC}\end{align*}

Prove: \begin{align*}\overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}\end{align*}

Statement Reason
1.
2. Definition of a parallelogram
3. Alternate Interior Angles Theorem
4. \begin{align*}\overline{AB} \cong \overline{DC}\end{align*}
5.
6. \begin{align*}\overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}\end{align*}
1. Challenge Find \begin{align*}x, y^\circ,\end{align*} and \begin{align*}z^\circ\end{align*}. (The two quadrilaterals with the same side are parallelograms.)

1. \begin{align*}3x+x+3x+x=360^\circ\!\\ {\;}\qquad \qquad \quad \ 8x = 360^\circ\!\\ {\;}\quad \ \qquad \quad \ \ \ \ x = 45^\circ\end{align*}
2. \begin{align*}4x+2=90^\circ\!\\ {\;}\quad \ 4x=88^\circ\!\\ {\;}\qquad x=22^\circ\end{align*}

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