geometry assignment state of the two triangles are congruent

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4.12: Congruent Triangles

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Verify congruency with SSS, SAS, RHS, and ASA

Applications of Congruent Triangles

Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs are congruent.

The following list summarizes the different criteria that can be used to show triangle congruence:

• AAS (Angle-Angle-Side): If two triangles have two pairs of congruent angles, and a non-common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two triangles have two pairs of congruent angles and the common side of the angles (the side between the congruent angles) in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.
• SAS (Side-Angle-Side): If two triangles have two pairs of congruent sides and the included angle in one triangle is congruent to the included angle in the other triangle, then the triangles are congruent.
• SSS (Side-Side-Side): If two triangles have three pairs of congruent sides, then the triangles are congruent.
• Right triangles only: HL (Hypotenuse-Leg): If two right triangles have one pair of legs congruent and hypotenuses congruent, then the triangles are congruent.

If two triangles don't satisfy at least one of the criteria above, you cannot be confident that they are congruent.

Interactive Element

Recognizing Perpendicular Bisectors

In the triangle below, \overline{BC} is the perpendicular bisector of AD\overline{AB}. Therefore \overline{AC}\cong \overline{CD}. Also, m\angle ACB=90^{\circ}and m\angle DCB=90^{\circ}, so \angle ACB \cong \angle DCB. You also know that \overline{BC} is a side of both triangles, and is clearly congruent to itself (this is called the reflexive property).

The triangles are congruent by SAS. Note that even though these are right triangles, you would not use HL to show triangle congruence in this case since you are not given that the hypotenuses are congruent.

Measuring Angles

Using the information from the previous problem, if \(m\angle A=50^{\circ}\), what is \(m\angle D\)?

\(m\angle D=50^{\circ}\)

Since the triangles are congruent, all of their corresponding angles and sides must be congruent. \angle A\) and \angle D\) are corresponding angles, so \(\angle A\cong \angle D\).

Congruent Triangles

Does one diagonal of a rectangle divide the rectangle into congruent triangles?

• Recall that a rectangle is a quadrilateral with four right angles.
• The opposite sides of a rectangle are congruent.

There is more than enough information to show that \(\Delta EFG\cong \Delta GHE)\.

• Method #1: The triangles have three pairs of congruent sides, so they are congruent by SSS.
• Method #2: The triangles have two pairs of congruent sides and congruent included angles, so they are congruent by SAS.
• Method #3: The triangles are right triangles with congruent hypotenuses and a pair of congruent legs, so they are congruent by HL.

Example \(\PageIndex{1}\)

Max constructs a triangle using an online tool. He tells Alicia that his triangle has a 42^{\circ} angle, a side of length 12 and a side of length 8. With only this information, will Alicia be able to construct a triangle that must be congruent to Max's triangle?

If Max also told Alicia that the angle was in between the two sides, then she would be able to construct a triangle that must be congruent due to SAS. If the angle is not between the two sides, she cannot be confident that her triangle is congruent because SSA is not a criterion for triangle congruence. Because Max did not state where the angle was in relation to the sides, Alicia cannot create a triangle that must be congruent to Max's triangle.

Example \(\PageIndex{2}\)

Are the following triangles congruent? Explain.

Notice that besides the one pair of congruent sides and the one pair of congruent angles, \(\overline{AC}\cong \overline{CA}\).

\(\Delta ACB\cong \Delta CAD\) by SAS.

Example \(\PageIndex{3}\)

The congruent sides are not corresponding in the same way that the congruent angles are corresponding. The given information for \(\Delta ACB\) is SAS while the given information for \(\Delta CAD\) is SSA. The triangles are not necessarily congruent.

Example \(\PageIndex{4}\)

\(G\) is the midpoint of \(\overline{EH}\). Are the following triangles congruent? Explain.

Because G\) is the midpoint of \(\overline{EH}\), \(\overline{EG}\cong \overline{GH}\). You also know that \(\angle EGF\cong \angle HGI\) because they are vertical angles. \(\Delta EGF\cong \Delta HGI\) by ASA.

1. List the five criteria for triangle congruence and draw a picture that demonstrates each.

2. Given two triangles, do you always need at least three pieces of information about each triangle in order to be able to state that the triangles are congruent?

For each pair of triangles, tell whether the given information is enough to show that the triangles are congruent. If the triangles are congruent, state the criterion that you used to determine the congruence and write a congruency statement.

Note that the images are not necessarily drawn to scale.

For 9-11, state whether the given information about a hidden triangle would be enough for you to construct a triangle that must be congruent to the hidden triangle. Explain your answer.

9. \(\Delta ABC\) with \(m\angle A=72^{\circ},\: AB=6 \:cm, \:BC=8 \:cm.\)

10. \(\Delta ABC\) with \(m\angle A=90^{\circ},\: AB=4 \:cm, \:BC=5 \:cm.\)

11. \(\Delta ABC\) with \(m\angle A=72^{\circ},\: AB=6 \:cm, \:AC=8 \:cm.\)

12. Recall that a square is a quadrilateral with four right angles and four congruent sides. Show and explain why a diagonal of a square divides the square into two congruent triangles.

13. Show and explain using a different criterion for triangle congruence why a diagonal of a square divides the square into two congruent triangles.

14. Recall that a kite is a quadrilateral with two pairs of adjacent, congruent sides. Will one of the diagonals of a kite divide the kite into two congruent triangles? Show and explain your answer.

15. In the picture below, \(G\) is the midpoint of both \(\overline{EH}\) and \(\overline{FI}\). Explain why \(\overline{FH}\cong \overline{IE}\) and \(\overline{FE}\cong \overline{HI}\).

16. Explain why AAA is not a criterion for triangle congruence.

Review (Answers)

To see the Review answers, click here.

Additional Resources

Video: Congruent and Similar Triangles - KA

Practice: Congruent Triangles

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2.1: The Congruence Statement

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• Henry Africk
• CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

Two triangles are said to be congruent if one can be placed over the other so that they coincide (fit together). This means that congruent triangles are exact copies of each other and when fitted together the sides and angles which coincide, called corresponding sides and angles, are equal.

In Figure \(\PageIndex{1}\), \(\triangle ABC\) is congruent to \(\triangle DEF\). The symbol for congruence is \(\cong\) and we write \(\triangle ABC \cong \triangle DEF\). \(\angle A\) corresponds to \(\angle D\), \(\angle B\) corresponds to \(\angle E\), and \(\angle C\) corresponds to \(\angle F\). Side \(AB\) corresponds to \(DE, BC\) corresponds to \(EF\), and \(AC\) corresponds to \(DF\).

In this book the congruence statement \(\triangle ABC \cong \triangle DEF\) will always be written so that corresponding vertices appear in the same order, For the triangles in Figure \(\PageIndex{1}\), we might also write \(\triangle BAC \cong \triangle EDF\) or \(\triangle ACB \cong \triangle DFE\) but never for example \(\triangle ABC \cong \triangle EDF\) nor \(\triangle ACB \cong \triangle DEF\). (Be warned that not all textbooks follow this practice, Many authors wil write the letters without regard to the order. If that is the case then we cannot tell which parts correspond from the congruence statement)

Therefore we can always tell which parts correspond just from the congruence statement. For example, given that \(\triangle ABC \cong \triangle DEF\), side \(AB\) corresponds to side \(DE\) because each consists of the first two letters, \(AC\) corresponds to DF because each consists of the first and last letters, \(BC\) corresponds to \(EF\) because each consists of the last two letters.

Example \(\PageIndex{1}\)

If \(\triangle PQR \cong \triangle STR\)

• list the corresponding angles and sides;
• find \(x\) and \(y\).

\(\begin{array} {rcll} {\underline{\triangle PQR}} & \ & {\underline{\triangle STR}} & {} \\ {\angle P} & = & {\angle S} & {\text{(first letter of each triangle in congruence statement)}} \\ {\angle Q} & = & {\angle T} & {\text{(second letter)}} \\ {\angle PRQ} & = & {\angle SRT} & {\text{(third letter. We don't write "}\angle R = \angle R \text{" since}} \\ {} & & {} & {\text{each }\angle R \text{ is different)}} \\ {PQ} & = & {ST} & {\text{(first two letters)}} \\ {PR} & = & {SR} & {\text{(firsst and last letters)}} \\ {QR} & = & {TR} & {\text{(last two letters)}} \end{array}\)

\(x = PQ = ST = 6\).

\(y = PR = SR = 8\).

Answer (2): \(x = 6, y = 8\).

Example \(\PageIndex{2}\)

Assuming \(\triangle I \cong \triangle II\), write a congruence statement for \(\triangle I\) and \(\triangle II\):

\(\begin{array} {rcll} {\triangle I} & \ & {\triangle II} & {} \\ {\angle A} & = & {\angle B} & {(\text{both = } 60^{\circ})} \\ {\angle ACD} & = & {\angle BCD} & {(\text{both = } 30^{\circ})} \\ {\angle ADC} & = & {\angle BDC} & {(\text{both = } 90^{\circ})} \end{array}\)

Answer: \(\triangle ACD \cong \triangle BCD\).

Example \(\PageIndex{3}\)

The angles that are marked the same way are assumed to be equal.

\(\begin{array} {rcll} {\underline{\triangle I}} & \ & {\underline{\triangle II}} & {} \\ {\angle A} & = & {\angle B} & {(\text{both marked with one stroke})} \\ {\angle ACD} & = & {\angle BCD} & {(\text{both marked with two strokes})} \\ {\angle ADC} & = & {\angle BDC} & {(\text{both marked with three strokes})} \end{array}\)

The relationships are the same as in Example \(\PageIndex{2}\).

Answer : \(\triangle ACD \cong \triangle BCD\).

1 - 4. For each pair of congruent triangles

(1) list the corresponding sides and angles;

(2) find \(x\) and \(y\).

1. \(\triangle ABC \cong \triangle DEF\).

2. \(\triangle PQR \cong \triangle STU\).

3. \(\triangle ABC \cong \triangle CDA\).

4. \(\triangle ABC \cong \triangle EDC\).

5 - 10. Write a congruence statement for each of the following. Assume the triangles are congruent and that angles or sides marked in the same way are equal.

How To Prove Triangles Congruent - SSS, SAS, ASA, AAS Rules

In these lessons, we will learn

• the SSS, SAS, ASA and AAS rules,
• how to use two-column proofs to prove triangles congruent.

Related Pages Congruent Triangles More Geometry Lessons

Congruent Triangles

Rules for triangle congruency.

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule. In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule . As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

The following diagrams show the Rules for Triangle Congruency: SSS, SAS, ASA, AAS and RHS. Take note that SSA is not sufficient for Triangle Congruency. Scroll down the page for more examples, solutions and proofs.

Side-Side-Side (SSS) Rule

Side-Side-Side is a rule used to prove whether a given set of triangles are congruent.

The SSS rule states that: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

In the diagrams below, if AB = RP , BC = PQ and CA = QR , then triangle ABC is congruent to triangle RPQ .

Side-Angle-Side (SAS) Rule

Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent.

The SAS rule states that: If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is an angle formed by two given sides.

Included Angle           Non-included angle

For the two triangles below, if AC = PQ , BC = PR and angle C< = angle P , then by the SAS rule, triangle ABC is congruent to triangle QRP .

Angle-Side-Angle (ASA) Rule

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

The ASA rule states that: If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Rule

The AAS rule states that: If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagrams below, if AC = QP , angle A = angle Q , and angle B = angle R , then triangle ABC is congruent to triangle QRP .

Three Ways To Prove Triangles Congruent

A video lesson on SAS, ASA and SSS.

• SSS Postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
• SAS Postulate: If there exists a correspondence between the vertices of two triangles such that the two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
• ASA Postulate: If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

Using Two Column Proofs To Prove Triangles Congruent

Triangle Congruence by SSS How to Prove Triangles Congruent using the Side Side Side Postulate? If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Triangle Congruence by SAS How to Prove Triangles Congruent using the SAS Postulate? If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Prove Triangle Congruence with ASA Postulate How to Prove Triangles Congruent using the Angle Side Angle Postulate? If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Prove Triangle Congruence by AAS Postulate How to Prove Triangles Congruent using the Angle Angle Side Postulate? If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

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How To Find if Triangles are Congruent

There are five ways to find if two triangles are congruent: SSS , SAS , ASA , AAS and HL .

1. SSS   (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

(See Solving SSS Triangles to find out more)

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS   (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

(See Solving SAS Triangles to find out more)

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA   (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

(See Solving ASA Triangles to find out more)

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS   (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

(See Solving AAS Triangles to find out more)

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL   (hypotenuse, leg)

This one applies only to right angled-triangles !

HL stands for " H ypotenuse, L eg" because the longest side of a right-angled triangle is called the "hypotenuse" and the other two sides are called "legs".

It means we have two right-angled triangles with

• the same length of hypotenuse and
• the same length for one of the other two legs .

It doesn't matter which leg since the triangles could be rotated.

(See Pythagoras' Theorem to find out more)

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes :

Without knowing at least one side, we can't be sure if two triangles are congruent.

Congruent Triangles

A very important topic in the study of geometry is congruence. Thus far, we have only learned about congruent angles, but in this section we will learn about the criteria necessary for triangles to be congruent. Learning about congruence on this level will open the door to different triangle congruence theorems that characterize geometry.

Corresponding Parts

Recall that in order for lines or angles to be congruent, they had to have equal measures. In that same way, congruent triangles are triangles with corresponding sides and angles that are congruent, giving them the same size and shape. Because side and angle correspondence is important, we have to be careful with the way we name triangles. For instance, if we have ?ABC??DEF , the congruence between triangles implies the following:

It is important to name triangles correctly to identify which segments are equal in length and to see which angles have the same measures.

In short, we say that two triangles are congruent if their corresponding parts (which include lines and angles) are congruent. In a two-column geometric proof , we could explain congruence between triangles by saying that “corresponding parts of congruent triangles are congruent.” This statement is rather long, however, so we can just write “CPCTC” for short.

Third Angles Theorem

In some instances we will need a very significant theorem to help us prove congruence between two triangles. If we know that two angles of two separate triangles are congruent, our inclination is to believe that their third angles are equal because of the Triangle Angle Sum Theorem .

This type of reasoning is correct and is a very helpful theorem to use when trying to prove congruence between triangles. The Third Angles Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent also.

Let’s take a look at some exercises to put our knowledge of congruent triangles, CPCTC, and the Third Angles Theorem to work.

(1) Which of the following expresses the correct congruence statement for the figure below?

While it may not seem important, the order in which you list the vertices of a triangle is very significant when trying to establish congruence between two triangles. Essentially what we want to do is find the answer that helps us correspond the triangles’ points, sides, and angles. The answer that corresponds these characteristics of the triangles is (b) .

In answer (b) , we see that ?PQR ? ?LJK . Let’s start off by comparing the vertices of the triangles. In the first triangle, the point P is listed first. This corresponds to the point L on the other triangle. We know that these points match up because congruent angles are shown at those points. Listed next in the first triangle is point Q . We compare this to point J of the second triangle. Again, these match up because the angles at those points are congruent. Finally, we look at the points R and K . The angles at those points are congruent as well.

We can also look at the sides of the triangles to see if they correspond. For instance, we could compare side PQ to side LJ . The figure indicates that those sides of the triangles are congruent. We can also look at two more pairs of sides to make sure that they correspond. Sides QR and JK have three tick marks each, which shows that they are congruent. Finally, sides RP and KJ are congruent in the figure. Thus, the correct congruence statement is shown in (b) .

(2) Find the values of x and y given that ?MAS ? ?NER .

We have two variables we need to solve for. It would be easiest to use the 16x to solve for x first (because it is a single-variable expression), as opposed to using the side NR , would require us to try to solve for x and y at the same time. We must look for the angle that correspond to ?E so we can set the measures equal to each other. The angle that corresponds to ?E is ?A , so we get

Now that we have solved for x , we must use it to help us solve for y . The side that RN corresponds to is SM , so we go through a similar process like we did before.

Now we substitute 7 for x to solve for y :

We have finished solving for the desired variables.

To begin this problem, we must be conscious of the information that has been given to us. We know that two pairs of sides are congruent and that one set of angles is congruent. In order to prove the congruence of ?RQT and ?SQT , we must show that the three pairs of sides and the three pairs of angles are congruent.

Since QS is shared by both triangles, we can use the Reflexive Property to show that the segment is congruent to itself. We have now proven congruence between the three pairs of sides. The congruence of the other two pairs of sides were already given to us, so we are done proving congruence between the sides.

Now we must show that all angles are congruent within the triangles. One pair has already been given to us, so we must show that the other two pairs are congruent. It has been given to us that QT bisects ?RQS . By the definition of an angle bisector, we know that two equivalent angles exist at vertex Q . The final pairs of angles are congruent by the Third Angles Theorem (since the other two pairs of corresponding angles of the triangles were congruent). We conclude that the triangles are congruent because corresponding parts of congruent triangles are congruent. The two-column geometric proof that shows our reasoning is below.

We are given that the three pairs of corresponding sides are congruent, so we do not have to worry about this part of the problem; we only need to worry about proving congruence between corresponding angles.

We are only given that one pair of corresponding angles is congruent, so we must determine a way to prove that the other two pairs of corresponding angles are congruent. We do this by showing that ?ACB and ?ECD are vertical angles. So, by the Vertical Angles Theorem , we know that they are congruent to each other. Now that we know that two of the three pairs of corresponding angles of the triangles are congruent, we can use the Third Angles Theorem . This theorem states that if we have two pairs of corresponding angles that are congruent, then the third pair must also be congruent.

Since all three pairs of sides and angles have been proven to be congruent, we know the two triangles are congruent by CPCTC . The two-column geometric proof that shows our reasoning is below.

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High school geometry

Course: high school geometry   >   unit 3.

• Triangle congruence postulates/criteria
• Determining congruent triangles
• Calculating angle measures to verify congruence
• Determine congruent triangles
• Corresponding parts of congruent triangles are congruent
• Proving triangle congruence

Prove triangle congruence

• Triangle congruence review