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What is Congruency?
In geometry, congruent triangles are triangles that have the same size and shape. Congruent triangles can be easily recognized because they can be superimposed on each other, with one triangle fitting exactly over the other.
There are several ways to prove that two triangles are congruent. One way is to use the Side-Side-Side (SSS) Congruence Theorem, which states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Another way is to use the Angle-Side-Angle (ASA) Congruence Theorem, which states that 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.
It is also possible to prove that two triangles are congruent using the Angle-Angle-Side (AAS) Congruence Theorem, which states that 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.
Finally, it is possible to prove that two triangles are congruent using the Hypotenuse-Leg (HL) Congruence Theorem, which states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
Knowing how to prove that two triangles are congruent is important in geometry, as it allows you to make conclusions about the size and shape of triangles based on limited information.
Congruent meaning in Maths
In mathematics, the word “congruent” means “having the same size and shape.” Two geometric figures are congruent if they can be superimposed on each other, with one figure fitting exactly over the other. For example, two triangles are congruent if they have the same size and shape, and two circles are congruent if they have the same size and shape.
The concept of congruence is used in many areas of mathematics, including geometry, trigonometry, and algebra. In geometry, it is often used to describe the relationship between geometric figures, such as triangles, circles, and squares. In trigonometry, it is used to describe the relationship between angles and side lengths in triangles. In algebra, it is used to describe the relationship between equations and their solutions.
The symbol for congruence is a tilde (~). It is used to indicate that two geometric figures are congruent, and is often written as follows:
Figure 1 ~ Figure 2
This notation reads as “Figure 1 is congruent to Figure 2.”
Knowing how to determine whether two figures are congruent is important in mathematics, as it allows you to make conclusions about the size and shape of figures based on limited information. It is also a useful tool for solving problems involving geometric figures.
Here is a diagram showing how to construct congruent triangles:
- Draw two triangles on a piece of paper.
- Choose one of the triangles as the reference triangle.
- Use a ruler to draw a line segment connecting a vertex of the reference triangle to a point on the other triangle. This line segment is called the “base.”
- Use a compass to draw an arc that passes through the vertex of the reference triangle and intersects the base.
- Without changing the compass width, place the compass on the point where the arc intersects the base and draw another arc on the other triangle.
- Use a straightedge to draw a line segment connecting the point where the second arc intersects the base to the vertex of the other triangle.
- Repeat steps 4 through 6 for the other two vertices of the reference triangle.
- The two triangles are congruent if the corresponding sides and angles of the two triangles are equal
Corresponding parts of congruent triangles are the parts of the two triangles that have the same size and position. For example, in the diagram below, the sides AB and PQ, the sides AC and PR, and the sides BC and QR are corresponding sides of the congruent triangles ABC and PQR. Similarly, the angles A and P, the angles B and Q, and the angles C and R are corresponding angles of the congruent triangles ABC and PQR.
[asy] pair A,B,C,P,Q,R;A = (0,0); B = (1,0); C = (0.5,0.866); P = (2,0); Q = (3,0); R = (2.5,0.866);
draw(A–B–C–cycle); draw(P–Q–R–cycle); draw(A–P); draw(B–Q); draw(C–R); label(“$A$”,A,SW); label(“$B$”,B,SE); label(“$C$”,C,N); label(“$P$”,P,SW); label(“$Q$”,Q,SE); label(“$R$”,R,N); [/asy]
In general, if two triangles are congruent, then their corresponding sides and angles are equal. For example, if triangle ABC is congruent to triangle PQR, then AB = PQ, AC = PR, and BC = QR. Similarly, angle A = angle P, angle B = angle Q, and angle C = angle R.
Knowing about corresponding parts of congruent triangles is important in geometry, as it allows you to make conclusions about the size and shape of triangles based on limited information. It is also a useful tool for solving problems involving geometric figures.
The Side-Side-Side (SSS) Congruence Theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. In other words, if two triangles have three sides of the same length, then they must be the same size and shape.
For example, consider the following two triangles:
[asy] pair A,B,C,P,Q,R;A = (0,0); B = (1,0); C = (0.5,0.866); P = (2,0); Q = (3,0); R = (2.5,0.866);
draw(A–B–C–cycle); draw(P–Q–R–cycle); label(“$A$”,A,SW); label(“$B$”,B,SE); label(“$C$”,C,N); label(“$P$”,P,SW); label(“$Q$”,Q,SE); label(“$R$”,R,N); label(“$3$”,(A+B)/2,S); label(“$4$”,(B+C)/2,E); label(“$5$”,(A+C)/2,N); label(“$3$”,(P+Q)/2,S); label(“$4$”,(Q+R)/2,E); label(“$5$”,(P+R)/2,N); [/asy]
In this example, the sides AB, BC, and AC of triangle ABC are all congruent to the sides PQ, QR, and PR of triangle PQR. Therefore, by the SSS Congruence Theorem, triangles ABC and PQR are congruent.
The SSS Congruence Theorem is often abbreviated as “SSS.” It is one of several ways to prove that two triangles are congruent, along with the Angle-Side-Angle (ASA) Congruence Theorem, the Angle-Angle-Side (AAS) Congruence Theorem, and the Hypotenuse-Leg (HL) Congruence Theorem.
Knowing how to use the SSS Congruence Theorem is important in geometry, as it allows you to make conclusions about the size and shape of triangles based on limited information. It is also a useful tool for solving problems involving geometric figures.
The Side-Angle-Side (SAS) Congruence Theorem states that 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. In other words, if two triangles have two sides of the same length and the included angle is the same size, then they must be the same size and shape.
For example, consider the following two triangles:
[asy] pair A,B,C,P,Q,R;A = (0,0); B = (1,0); C = (0.5,0.866); P = (2,0); Q = (3,0); R = (2.5,0.866);
draw(A–B–C–cycle); draw(P–Q–R–cycle); draw(A–P); label(“$A$”,A,SW); label(“$B$”,B,SE); label(“$C$”,C,N); label(“$P$”,P,SW); label(“$Q$”,Q,SE); label(“$R$”,R,N); label(“$3$”,(A+B)/2,S); label(“$4$”,(B+C)/2,E); label(“$5$”,(A+C)/2,N); label(“$3$”,(P+Q)/2,S); label(“$4$”,(Q+R)/2,E); label(“$5$”,(P+R)/2,N); label(“$\angle A$”,A,2W); label(“$\angle P$”,P,2W); [/asy]
In this example, the sides AB and AC of triangle ABC are congruent to the sides PQ and PR of triangle PQR, and angle A is congruent to angle P. Therefore, by the SAS Congruence Theorem, triangles ABC and PQR are congruent.
The SAS Congruence Theorem is often abbreviated as “SAS.” It is one of several ways to prove that two triangles are congruent, along with the Side-Side-Side (SSS) Congruence Theorem, the Angle-Angle-Side (AAS) Congruence Theorem, and the Hypotenuse-Leg (HL) Congruence Theorem.
Knowing how to use the SAS Congruence Theorem is important in geometry, as it allows you to make conclusions about the size and shape of triangles based on limited information. It is also a useful tool for solving problems involving geometric figures.
The Angle-Angle-Side (AAS) Congruence Theorem is a variation of the Angle-Side-Angle (ASA) Congruence Theorem. It states that 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. In other words, if two triangles have two angles of the same size and a side that is not included between the two angles is the same length, then they must be the same size and shape.
The AAS Congruence Theorem can be proved using the ASA Congruence Theorem as follows:
- Draw the non-included side of one of the triangles.
- Draw the included side that is opposite to one of the angles.
- Draw the included side that is opposite to the other angle.
- Use the ASA Congruence Theorem to prove that the two triangles are congruent.
For example, consider the following two triangles:
[asy] pair A,B,C,P,Q,R;A = (0,0); B = (1,0); C = (0.5,0.866); P = (2,0); Q = (3,0); R = (2.5,0.866);
draw(A–B–C–cycle); draw(P–Q–R–cycle); draw(A–P); draw(B–Q); draw(C–R); label(“$A$”,A,SW); label(“$B$”,B,SE); label(“$C$”,C,N); label(“$P$”,P,SW); label(“$Q$”,Q,SE); label(“$R$”,R,N); label(“$3$”,(A+B)/2,S); label(“$4$”,(B+C)/2,E); label(“$5$”,(A+C)/2,N); label(“$3$”,(P+Q)/2,S); label(“$4$”,(Q+R)/2,E); label(“$5$”,(P+R)/2,N); label(“$\angle A$”,A,2W); label(“$\angle P$”,P,2W); label(“$\angle B$”,B,2E); label(“$\angle Q$”,Q,2E); label(“$\angle C$”,C,2N); label(“$\angle R$”,R,2N); [/asy]
In this example, the angles A and C of triangle ABC are congruent to the angles P and R of triangle PQR, and the side AC is congruent to the side PR. Therefore, we can use the AAS Congruence Theorem to prove that triangles ABC and PQR are congruent.
The AAS Congruence Theorem is often abbreviated as “AAS.” It is one of several ways to prove that two triangles are congruent, along with the Side-Side-Side (SSS) Congruence Theorem, the Angle-Side-Angle (ASA) Congruence Theorem, and the Hypotenuse-Leg (HL) Congruence Theorem.
