Topics
7. Polygons
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United States of AmericaVirginia
Grade 6

7.01 Congruent figures

Lesson
Practice

Congruent segments

Segment

A segment starts at one point and stops at the other.

Segment xy
Segment AB

This segment can be named using its endpoints. We call this \overline{AB}. The line over the letters means segment, so we say "segment AB".

We can also use the reverse order to name the same segment. \overline{BA} or "segment BA".

When talking about the length of the segment or the distance between points A and B we can say AB without the line over top.

It's important that when discussing a segment, we use the line over the two letters to indicate segment. For example, \overline{EF} or \overline{FE} is a segment, while EF or FE is a distance or length.

Exploration

Move the AB and CD sliders to change the lengths of the segments.

Move the "Overlap" slider to move segment AB so that it is directly on top of segment CD.

Loading interactive...

  1. Set the sliders for each segment to be the same number then overlap them. What do you notice?

  2. Set the sliders for each segment to be different numbers then overlap them. What do you notice?

  3. Repeat both with new numbers. Do you notice the same things each time?

Congruent segments

Line segments that have the same length. The symbol \cong is used to represent congruence.

We place small markings on segments when we want to show that they are equal in length.

Segments AC and AB are connected through point A. Each segment has one-stroke markings and labeled '2 in'.

The small identical markings on each segment, called 'hash' or 'hatch' marks, tell us they are equal in length, or congruent.

We also see \overline{AB} has the same length, 2 \text{ in} as \overline{AC}. We can say AB=AC when discussing length, but we must say segment \overline{AB} is congruent to segment \overline{AC}.

We can write a congruency statement using our congruence symbol:

\overline{AB}\cong\overline{AC}

Lengths and distances are said to be equal while segments are congruent.

This does not mean that the two segments are made up of the same points - only that they have the same length. Sometimes we will use more than one kind of marking to show that some segments are equal to others.

In this diagram we use both single hatch markings and double hatch markings. The segments with single hatch markings are congruent to each other. The segments with double hatch markings are congruent to each other.

Segments AB and BC have two-stroke markings while segments CD and DE have one-stroke markings. AB and BC are labeled '1 in', and CD and DE are labeled '2 in'.

The length of AB is equal to the length of BC, so \overline{AB} is congruent to \overline{BC}.

\overline{AB}\cong\overline{BC}

The length of CD is equal to the length of DE, so \overline{CD} is congruent to \overline{DE}.

\overline{CD}\cong\overline{DE}

Examples

Example 1

Use the diagram below to answer the following:

Segments XW, ZX, ZY and XY. Segments XW and ZY have one-stroke markings.
a

What segment is congruent to \overline{ZY}?

Worked Solution
Create a strategy

Choose the segment with the same marking as \overline{ZY}.

Apply the idea

Segment \overline{XW} has the same marking as \overline{ZY}.

This shows us that \overline{XW} is congruent to \overline{ZY}.

Reflect and check

Since segments are named using their endpoints, the order of letters does not matter. We could have also said segment \overline{WX} is congruent to \overline{ZY}.

b

Write a congruence statement

Worked Solution
Create a strategy

Use the congruence symbol \cong to write a conguence statement for the two segments.

Apply the idea

\overline{XW}\cong \overline{ZY}

Reflect and check

The congruence statement \overline{WX}\cong \overline{ZY} is also true.

Example 2

Identify any congruent segments in the figure below and write a congruence statement for each.

Segment AB 3ft, segment BC 4ft, segment CA 4ft and segment CD 3ft. segments AB, BC, and CD forms a triangle.
Worked Solution
Create a strategy

Remember, congruent segments are equal in length.

Apply the idea

Since the length of \overline{AC} is equal to the length of \overline{BC}, we can say \overline{AC} \cong \overline{BC}.

Since the length of \overline{AB} is equal to the length of \overline{CD}, we can say \overline{AB} \cong \overline{CD}.

Idea summary
Segments AB and BC have two-stroke markings while segments CD and DE have one-stroke markings. AB and BC are labeled '1 in', and CD and DE are labeled '2 in'.
  • Congruent segments are equal in length.
  • We place small markings to show the segments that are equal in length.
  • We use our congruency symbol \cong to write congruency statements. \\\overline{AB}\cong\overline{BC} and \overline{CD}\cong\overline{DE}

Congruent angles

Angle

An angle is formed when two rays, lines, or segments, are joined at their endpoints. Angles are measured in degrees.

Two rays that share an endpoint.

Whenever two lines, rays, or segments pass through the same point, we can describe the amount of turn or rotation it takes to get from one to the other using an angle.

Rays AB and AC drawn from the same point A. the two rays forms an 85degree angle. Point A is labeled 'vertex'.

\, \\\\Here are two rays drawn from the same point forming an angle.

The vertex of an angle is the angle formed by two lines or rays that intersect at a point.

In this image, our vertex angle is \angle A. We use the letter at the vertex to name the angle.

Triangle XYZ with angle measurements of 75 , 40 and 65 degrees.

Let's name the three angles of the triangle:

  • \angle X

  • \angle Y

  • \angle Z

Now let's identify the measure of each angle:

  • m\angle X=75\degree

  • m\angle Y=40\degree

  • m\angle Z=65\degree

Exploration

Move the Angle A and Angle B sliders to change the size of the angles.

Move the "Overlap" slider to move Angle A so that it is directly on top of Angle B.

Loading interactive...

  1. Set the sliders for each angle to be the same degree measure then overlap them. What do you notice?

  2. Set the sliders for each angle to be different degree measures then overlap them. What do you notice?

  3. Repeat both with new numbers. Do you notice the same things each time?

Similar to segments, we must be careful with our notation when discussing angles. Congruent angles have the same angle measure.

Angles B and Q, both at 40degree angles but angle Q is slanted.

Since the measure of \angle B is equal to the measure of \angle Q, we know \angle B is congruent to \angle Q.

Congruence Statement:

\angle B \cong \angle Q

Since \angle B and \angle Q have the same angle marking, this also tells us they are equal in measure and therefore congruent.

Just like with segments, we can use additional markings to show that two angles are congruent. We draw multiple arcs to show that different angles are congruent to each other.

Segments AB and AC forming angle 1. Segments FC and CE forming angle 3. Segments AC and FC are forming angle 2. Segments EC and CD are forming angle 4.

In this diagram the two angles drawn with double arcs have equal measure. Since the measure of the angles is equal, we know the angles are congruent.

Congruence Statement:

\angle 1 \cong\angle 4

There are no angles that are congruent to \angle 2 and \angle 3 because there are no other angles with the same number of markings.

Examples

Example 3

Use the image to answer the questions.

Two angles BCD abd EAD, sharing the same point D.
a

Which angles are congruent?

Worked Solution
Create a strategy

Look for angles with identitcal angle markings showing the angle measures are equal.

Apply the idea

The two angles marked as having equal measure are \angle C and \angle A. So we know \angle C is congruent to \angle A.

b

Write a congruence statement

Worked Solution
Create a strategy

Use the congruence symbol \cong to write a congruence statement for the angles we identified as congruent in part (a).

Apply the idea

\angle C \cong \angle A

Reflect and check

The congrunce statement \angle A \cong \angle C is also true.

Example 4

Use the image to answer the questions.

Tree angles S, G and W. Angle S has it rays point downward, and is 42 degrees. Angle G measures 42 degrees and angle W measures 34 degrees, both are facing right.
a

Which angles are congruent?

Worked Solution
Create a strategy

Identify any angles that have the same degree measure.

Apply the idea

The two angles with an equal measure are \angle G and \angle S since they both measure 42\degree. So we know that \angle G is congruent to \angle S.

b

Write a congruence statement

Worked Solution
Create a strategy

Use the congruence symbol \cong to write a congruence statement for the angles we identified as congruent in part (a).

Apply the idea

\angle G \cong \angle S

Reflect and check

The congruence statement \angle S \cong \angle G is also true.

Idea summary
Two angles B and Q both measuring 40 degrees. Angle Q is slanted.

  • Congruent angles have the same measure.

  • Congruent angles are marked using an arc or multiple arcs to show congruece

  • An angle is named using \angle symbol followed by letter of vertex angle. We use our congruence symbol, \cong to write a congruence statement. \angle B \cong \angle Q means angle B is congruent to angle Q.

Congruent polygons

Polygon

A polygon is a closed plane figure composed of at least three line segments that do not cross.

When we are naming a polygon, we use the labels on its vertices.

Triangle ABC
\triangle ABC
Rectangle EFGH
Rectangle EFGH

Exploration

Move the "Overlap" slider to move triangle DEF so it overlaps with triangle ABC.

Use the checkbox to show the side and anlge measures.

Click 'New figure' to get a new pair of triangles.

Loading interactive...

  1. Overlap several different pairs of triangles. What do you notice about the triangles that overlap perfectly?

  2. What do you notice about the triangles that do not overlap perfectly?

Congruent polygons

Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are congruent.

Corresponding angles are a pair of matching angles that are in the same spot in two different shapes. Corresponding sides are a pair of matching sides that are in the same spot in two different shapes.

Congruent triangles ABC and LMN. Angles A and M are 60 degrees, angles B and L are 50 degrees, and angles C and N are 70 degrees. side AB and side LM are both 3in, sides AC and MN are both 2 in, and sides BC and LN are both 4 in.

Here we have an image with two triangles.

Since the corresponding angles and corresponding sides are equal, we know these two triangles are congruent.

Two polygons with corresponding congruent angles, but the sides ae not congruent. Side measurements of the first polygon is 1m,3m,2m and 5m, while the second polygon has side measurements of 2m, 6m, 4m and 10m.

Here we have an image of two polygons.

The corresponding angles are congruent, however, the corresponding sides are not congruent.

This means that these two polygons are not congruent.

If two polygons are congruent, we can show that with a congruence statement. When writing congruence statements for polygons, the letters must be in the correct order according to the corresponding angles and sides.

Here is an example of two congruent triangles. Let's identify our corresponding angles and sides.

Congruent triangles GSP and RKF. Corresponding congruent angles are: G and R, S and K, P and F. Correponding congruent sides are: GS amd RK both at 10 units, SP and KF both at 5 units, PG and FR both at 8 units.

For the angles, the markings on image tell us:

  • \angle G\cong\angle R

  • \angle S\cong\angle K

  • \angle P\cong\angle F

For the sides, we see from the labeled lengths that:

  • \overline{GP}\cong\overline{RF}

  • \overline{GS}\cong\overline{RK}

  • \overline{SP}\cong\overline{KF}

This shows us that vertex G corresponds with vertex R,\,S with K and P with F so we can write the congruence satement:

\triangle GSP\cong\triangle RKF

The statements \triangle SPG\cong\triangle KFR and \triangle GPS\cong\triangle RFK are also true because they match the corresponding angles. There are several more true congruence statements we could write as long as we make sure the order matches up the congruent angles.

Examples

Example 5

Determine if the two polygons are congruent.

Triangles EFG and LMN. Congruent angles are EL, FM and GN. Congruent sides are EF and LM, EG and LM, FG and MN.
Worked Solution
Create a strategy

Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent. We see these triangles have an equal number of sides. So we need to check all corresponding sides and angles to determine if they are congruent.

Apply the idea

Let's first start by identifying the corresponding angles.

One arc marking is used with \angle E and \angle L, which tells us \angle E \cong \angle L. A double arc marking is used for \angle F and \angle M, showing that \angle F \cong \angle M. A triple arc marking is used for \angle G and \angle N, telling us that \angle G \cong \angle N.

All corresponding angles are congruent. Now let's check the corresponding sides.

Side EF is marked with one hatch mark and so is side LM. This tells us \overline{EF}\cong\overline{LM}. Both side EG and side LN are marked with two hatch marks, showing us \overline{EG}\cong\overline{LN}. Three hatch marks are used for side FG and side MN, showing us \overline{FG}\cong\overline{MN}.

Since all corresponding sides and angles are congruent, these two polygons are congruent.

Reflect and check

We can write a congruency statement:

\triangle EFG \cong \triangle LMN

Example 6

Determine if the two polygons are congruent.

Two rectangles JIHG and OPQR. The following sides measure 26mi: JI, HG, OP and QR. The following sides measure 44mi: IH, GJ, PQ and RO.
Worked Solution
Create a strategy

Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent. These polygons both have 4 sides, so we need to check all corresponding sides and angles to determine if they are congruent.

Apply the idea

Since all of the angles are marked as right \left(90 \degree \right) angles, we know all corresponding angles are congruent.

Let's check if the corresponding sides are congruent. Side JI and side OP, are both 26 \text{ mi} long, so \overline{JI}\cong\overline{OP}. Side JG in the first figure has a length of 44 miles and has no corresponding side of equal length in the second figure.

These two polygons are not congruent.

Reflect and check

Notice that we did not need to check all pairs of corresponding sides. As soon as we found one side that did not have a corresponding side of equal length we were able to say the figures are not congruent.

Idea summary
Congruent triangles ABC and LMN. Angles A and M are 60 degrees, angles B and L are 50 degrees, and angles C and N are 70 degrees. side AB and side LM are both 3in, sides AC and MN are both 2 in, and sides BC and LN are both 4 in.

Congruent polygons: Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are congruent.

When writing a congruency statement, we must make sure we are putting the letters in the correct order according to corresponding angles and sides.

Outcomes

6.MG.4

The student will determine congruence of segments, angles, and polygons.

6.MG.4c

Determine the congruence of segments, angles, and polygons given their properties.

6.MG.4d

Determine whether polygons are congruent or noncongruent according to the measures of their sides and angles.

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