assignment 1 defining congruent triangles

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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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1.10: Congruent triangles

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• Anton Petrunin
• Pennsylvannia State University

Our next goal is to give a rigorous meaning for (iv) on Section 1.1 . To do this, we introduce the notion of congruent triangles so instead of “if we rotate or shift we will not see the difference” we say that for triangles, the side-angle-side congruence holds; that is, two triangles are congruent if they have two pairs of equal sides and the same angle measure between these sides.

An ordered triple of distinct points in a metric space \(\mathcal{X}\), say \(A, B, C\), is called a triangle \(ABC\) (briefly \(\triangle ABC\)). Note that the triangles \(ABC\) and \(ACB\) are considered as different.

Two triangles \(A'B'C'\) and \(ABC\) are called congruent (it can be written as \(\triangle A'B'C' \cong \triangle ABC\)) if there is a motion \(f: \mathcal{X} \to \mathcal{X}\) such that

\(A' = f(A)\), \(B' = f(B)\) and \(C' = f(C)\).

Let \(\mathcal{X}\) be a metric space, and \(f, g: \mathcal{X} \to \mathcal{X}\) be two motions. Note that the inverse \(f^{-1}: \mathcal{X} \to \mathcal{X}\), as well as the composition \(f \circ g: \mathcal{X} \to \mathcal{X}\) are also motions.

It follows that "\(\cong\)" is an equivalence relation; that is, any triangle congruent to itself, and the following two conditions hold:

• If \(\triangle A'B'C' \cong \triangle ABC\), then \(\triangle ABC \cong \triangle A'B'C'\).
• If \(\triangle A''B''C'' \cong \triangle A'B'C'\) and \(\triangle A'B'C' \cong \triangle ABC\), then \[\triangle A''B''C'' \cong \triangle ABC.\]

Note that if \(\triangle A'B'C' \cong \triangle ABC\), then \(AB = A'B', BC = B'C'\) and \(CA = C'A'\).

For a discrete metric, as well as some other metrics, the converse also holds. The following example shows that it does not hold in the Manhattan plane:

Example \(\PageIndex{1}\)

Consider three points \(A = (0, 1), B = (1, 0)\), and \(C = (-1, 0)\) on the Manhattan plane \((\mathbb{R}^2, d_1)\). Note that

\[d_1 (A, B) = d_1 (A, C) = d_1 (B, C) = 2.\]

On one hand,

\(\triangle ABC \cong \triangle ACB.\)

Indeed, the map \((x, y) \mapsto (-x, y)\) is a motion of \((\mathbb{R}^2, d_1)\) that sends \(A \mapsto A, B \mapsto C\), and \(C \mapsto B\).

On the other hand,

\(\triangle ABC \not\cong \triangle BCA.\)

Indeed, arguing by contradiction, assume that \(\triangle ABC \cong \triangle BCA\); that is, there is a motion \(f\) of \((\mathbb{R}^2, d_1)\) that send \(A \mapsto B, B \mapsto C,\) and \(C \mapsto A\).

We say that \(M\) is a midpoint of \(A\) and \(B\) if

\(d_1(A, M) = d_1(B, M) = \dfrac{1}{2} \cdot d_1(A, B).\)

Note that a point \(M\) is a midpoint of \(A\) and \(B\) if and only if \(f(M)\) is a midpoint of \(B\) and \(C\).

The set of midpoints for \(A\) and \(B\) is infinite, it contains all points \((t, t)\) for \(t \in [0, 1]\) (it is the gray segment on the picture above). On the other hand, the midpoint for \(B\) and \(C\) is unique (it is the black point on the picture). Thus, the map \(f\) cannot be bijective — a contradiction.

• Math Article
• Congruence Of Triangles

Congruence of Triangles

Congruence of triangles: Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure. These triangles can be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with each other. The symbol of congruence is’ ≅’.

T he meaning of congruence in Maths is when two figures are similar to each other based on their shape and size. There are basically four congruence rules that proves if two triangles are congruent. But it is necessary to find all six dimensions. Hence, the congruence of triangles can be evaluated by knowing only three values out of six.  The corresponding sides and angles of congruent triangles are equal. Also, learn about  Congruent Figures  here.

Congruence is the term used to define an object and its mirror image. Two objects or shapes are said to be congruent if they superimpose on each other. Their shape and dimensions are the same. In the case of geometric figures, line segments with the same length are congruent and angle with the same measure are congruent.

CPCT is the term, we come across when we learn about the congruent triangle. Let’s see the condition for triangles to be congruent with proof.

Congruent meaning in Maths

The meaning of congruent in Maths is addressed to those figures and shapes that can be repositioned or flipped to coincide with the other shapes. These shapes can be reflected to coincide with similar shapes.

Two shapes are congruent if they have the same shape and size. We can also say if two shapes are congruent, then the mirror image of one shape is the same as the other.

Congruent Triangles

A closed polygon made of three line segments forming three angles is known as a Triangle.

Two triangles are said to be congruent if their sides have the same length and angles have same measure. Thus, two triangles can be superimposed side to side and angle to angle.

In the above figure, Δ ABC and Δ PQR are congruent triangles. This means,

Vertices:  A and P, B and Q, and C and R are the same.

Sides:  AB=PQ, QR= BC and AC=PR;

Angles:  ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.

Congruent triangles are triangles having corresponding sides and angles to be equal. Congruence is denoted by the symbol “≅”.   From the above example, we can write ABC ≅ PQR. They have the same area and the same perimeter.

For More Information On Introduction To Congruent Triangles, Watch The Below Video:

CPCT Full Form

CPCT is the term we come across when we learn about the congruent triangle. CPCT means “Corresponding Parts of Congruent Triangles”. As we know that the corresponding parts of congruent triangles are equal. While dealing with the concepts related to triangles and solving questions, we often make use of the abbreviation cpct in short words instead of full form.

CPCT Rules in Maths

The full form of CPCT is Corresponding parts of Congruent triangles. After proving triangles congruent, the remaining dimension can be predicted without actually measuring the sides and angles of a triangle. Different rules of congruency are as follows.

SSS (Side-Side-Side)

Sas (side-angle-side).

• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• RHS (Right angle-Hypotenuse-Side)

Let us learn them all in detail.

If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule.

In the above-given figure, AB= PQ, BC = QR and AC=PR, hence Δ ABC ≅ Δ PQR.

If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.

In above given figure, sides AB= PQ, AC=PR and angle between AC and AB equal to angle between PR and PQ i.e. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.

ASA (Angle-Side- Angle)

If any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.

In above given figure, ∠ B = ∠ Q, ∠ C = ∠ R and sides between ∠B and ∠C , ∠Q and ∠ R are equal to each other i.e. BC= QR. Hence, Δ ABC ≅ Δ PQR.

For More Information On SAS And ASA Congruency Rules, Watch The Below Video:

AAS (Angle-Angle-Side) [Application of ASA]

AAS stands for Angle-Angle-Side. When two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.

AAS congruence can be proved in easy steps.

Suppose we have two triangles ABC and DEF, where,

By angle sum property of triangle, we know that; ∠A + ∠B + ∠C = 180 ………(1) ∠D + ∠E + ∠F = 180 ……….(2)

From equation 1 and 2 we can say; ∠A + ∠B + ∠C = ∠D + ∠E + ∠F ∠A + ∠E + ∠F = ∠D + ∠E + ∠F [Since, ∠B = ∠E and ∠C = ∠F] ∠A = ∠D Hence, in triangle ABC and DEF, ∠A = ∠D AC = DF ∠C = ∠F Hence, by ASA congruency, Δ ABC ≅ Δ DEF

RHS (Right angle- Hypotenuse-Side)

If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent by RHS rule.

In above figure, hypotenuse XZ = RT and side YZ=ST, hence   ∆ XYZ ≅ ∆ RST.

Solved Example

Practice problems.

Q.1: PQR is a triangle in which PQ = PR and is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT. Q.2: If perpendiculars from any point within an angle on its arms are congruent. Prove that it lies on the bisector of that angle.

Video Lesson

Frequently Asked Questions

What are congruent triangles.

Two triangles are said to be congruent if the three sides and the three angles of both the angles are equal in any orientation.

What is the Full Form of CPCT?

CPCT stands for Corresponding parts of Congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.

What are the Rules of Congruency?

There are 5 main rules of congruency for triangles:

• SSS Criterion: Side-Side-Side
• SAS Criterion: Side-Angle-Side
• ASA Criterion: Angle-Side- Angle
• AAS Criterion: Angle-Angle-Side
• RHS Criterion: Right angle- Hypotenuse-Side

What is SSS congruency of triangles?

What is sas congruence of triangles, what is asa congruency of triangles, what is aas congruency, what is rhs congruency, leave a comment cancel reply.

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What is the difference between ASA rule and AAS rule ?

AAS Congruence Rule: Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. ASA Congruence Rule: Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.

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High school geometry

Course: high school geometry   >   unit 3.

• Quiz 1 Congruence