Topics
6. Polygons
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United States of AmericaVirginia
Grade 7

6.06 Find unknown measures in quadrilaterals

Lesson
Practice

Find unknown lengths in quadrilaterals

We can use the properties to find unknown lengths in a quadrilateral . Recall the properties of quadrilaterals related to length are:

ParallelogramRectangleRhombusSquareTrapezoidIsosceles Trapezoid
\text{All sides }\\\text{congruent}\checkmark\checkmark
\text{Opposite}\\\text{sides are}\\\text{congruent}\checkmark\checkmark\checkmark\checkmark
\text{Opposite}\\\text{sides parallel}\checkmark\checkmark\checkmark\checkmark\checkmark \text{(one pair)}\checkmark \text{(one pair)}
\text{Diagonals}\\\text{bisect each}\\\text{other}\checkmark\checkmark\checkmark\checkmark
\text{Diagonals are}\\\text{congruent}\checkmark\checkmark\checkmark\checkmark \text{(one pair)}
\text{Diagonals are}\\\text{perpendicular}\checkmark\checkmark

Examples

Example 1

Consider the rectangle ABCD below, where AC=16\text{ m} and AD=9\text{ m}.

Rectangle A B C D with two diagonals that bisect each other at right angles
a

Find BD.

Worked Solution
Create a strategy

Recall that the diagonals of a rectangle are equal in length.

Apply the idea

Since AC=BD and AC=16\text{ m}, then BD=16 \text{ m}

Reflect and check

Not only are the diagonals of a rectangle congruent, they also bisect each other. Since we know BD=16 \text{ m}, each half of the diagonal will have a length of 8\text{ m}.

b

Find BC.

Worked Solution
Create a strategy

Recall that opposite sides of a rectangle are equal in length.

Apply the idea

Since AD=BC, then BC=9 \text{ m}

Reflect and check

Opposite sides of a rectangle are always congruent, so we also know that AB=DC

Example 2

Consider the rhombus ABCD given below.

Rhombus ABCD with diagonals and center O. Side AD measures 5 centimeters, and DO measures 4 centimeters.
a

Find AB.

Worked Solution
Create a strategy

Recall that the sides of a rhombus are equal in length.

Apply the idea

Since AD=AB, then AB=5 \text{ cm}

b

Find the length of BD.

Worked Solution
Create a strategy

Recall that the diagonals of a rhombus bisect each other. This means, each diagonal cuts the other into two equal parts.

Apply the idea

Since \overline{BD} is cut in half by \overline{AC}, and DO=4\text{ cm}, then it must be true that BO=4\text{ cm}. We can find the entire length of \overline{BD} by adding together the two equal halves.

BD =4 \text{ cm} + 4\text{ cm}

BD =8 \text{ cm}

Reflect and check

Since the diagonals of a rhombus bisect each other, another method we could have used is to double the given half of diagonal to find the entire diagonal length.

Idea summary

The properties of the side and diagonal lengths of quadrilaterals are:

Table of properties of the side and diagonal lengths of quadrilaterals. Ask your teacher for more information.

Find unknown angles in quadrilaterals

One thing that all quadrilaterals have in common is that they can always be split down the middle to make two triangles. Since the sum of the angle measures in a triangle is 180 \degree, the angle sum of a quadrilateral is twice that: 360\degree.

We can see that illustrated in this diagram. The angles from each of the triangles form a straight angle, and together they form a full revolution.

This image shows the angle sum of a quadrilateral. Ask your teacher for more information.

This fact, along with the other properties of quadrilaterals, can be applied to solve for unknown measures in quadrilaterals.

ParallelogramRectangleRhombusSquareTrapezoidIsosceles Trapezoid
\text{All sides }\\\text{congruent}\checkmark\checkmark
\text{Opposite}\\\text{angles are}\\\text{congruent}\checkmark\checkmark\checkmark\checkmark
\text{All angles are}\\\text{right angle}\checkmark\checkmark
\text{Each diagonal}\\\text{bisect opposite}\\\text{angles}\checkmark\checkmark
\text{Base angles} \\ \text{are congruent}\checkmark

Examples

Example 3

Solve for the value of x in the diagram below.

A quadrilateral with angles of 104, 127, x, and 61 degrees.
Worked Solution
Create a strategy

To find the value of x, add all the angles and equate them to 360 degrees.

Apply the idea
\displaystyle x+104+127+61\displaystyle =\displaystyle 360Add the angles and set equal to 360
\displaystyle x+292\displaystyle =\displaystyle 360Evaluate the addition
\displaystyle x+292-292\displaystyle =\displaystyle 360-292Subtract 292 from both sides
\displaystyle x\displaystyle =\displaystyle 68Evalutate the subtraction

Example 4

Consider the rhombus PQRS below.

Rhombus P Q R S with two diagonals that bisect each other at right angles

What is the value of x?

Worked Solution
Create a strategy

Remember, the diagonals of a rhombus bisect each other at right angles.

Apply the idea

Since the diagonals are perpendicular, they form 90 \degree angles so: x=90

Example 5

Consider the rectangle ABCD below, find m\angle C.

A rectangle A B C D.
Worked Solution
Create a strategy

Remember, a rectangle has four right angles.

Apply the idea

Since \angle C is a right angle, it measures 90\degree.

Idea summary

The properties of angles of quadrilaterals are:

A table of properties of angles of quadrilaterals. Ask your teacher for more infomation.

Outcomes

7.MG.3

The student will compare and contrast quadrilaterals based on their properties and determine unknown side lengths and angle measures of quadrilaterals.

7.MG.3c

Given a diagram, determine an unknown angle measure in a quadrilateral, using properties of quadrilaterals.

7.MG.3d

Given a diagram, determine an unknown side length in a quadrilateral using properties of quadrilaterals.

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