Congruent shapes
If two shapes are congruent, they are identical in both shape and size.
Note: Shapes can be congruent even if one of them has been rotated or reflected.
Two triangles are congruent if they have:
• precisely the same three sides and • precisely the same three angles.
But we don’t have to know all three sides and all three angles, normally three out of the six is sufficient.
How To Find if Triangles are Congruent
There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.
Two triangles are congruent if one of the following conditions applies:
1. Three sides are the same
The three sides of the first triangle are equal to the three sides of the second triangle. This case is known as the SSS rule: Side Side Side. SSS stands for “side, side, side” and means that we have two triangles with all three sides equal.
For example:
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
2. Two sides and one angle are the same
Two sides of the first triangle are equal to two sides of the second triangle, and the included angle is equal. This is referred to as the SAS rule: 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.
For example:
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. Two angles and one side are the same
Two angles in the first triangle are equal to two angles in the second triangle, and one (equivalently situated) side is equal. This is referred to as the AAS rule: 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.
For example:
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.
4. Two sides in right-angled triangle are the same
In a right-angled triangle, the hypotenuse and one other side in the first triangle are equal to the hypotenuse and equivalent side in the second triangle. This is referred to as the RHS rule: Right angled, Hypotenuse, Side. It can also be referred to as HL (hypotenuse, leg)
This one applies only to right angled triangles!
Or
HL stands for “Hypotenuse, Leg” (the longest side of a right-angled triangle is known as the “hypotenuse”, the other two sides are known as “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.
For instance:
If the hypotenuse and one leg of one right-angled triangle are equal to the equivalent hypotenuse and leg of another right-angled triangle, the two triangles are congruent.
5. 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.
For example:
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.
Warning! Don’t make Use of “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 the knowledge of at least one side, we can’t be sure if two triangles are congruent.
Definition of Congruent Triangles
Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
In the simple case below, the two triangles PQR and LMN are congruent. This is so because everyone of the corresponding side has the same length, and every corresponding angle has the same measure. The angle at P has the same measure (in degrees) as the angle at L, the side PQ is the same length as the side LM etc.
In the diagram above, the triangles are drawn next to each other and it is clear that they are identical. Nevertheless, one triangle may be rotated, flipped over (reflected), or the two triangles may share a common side. These cases are discussed further on other pages:
• Rotated congruent triangles
• Reflected congruent triangles
• Congruent triangles with a common side
Assuming that the triangles are cardboard
One way to think about congruence triangle is to assume that they are made of cardboard. They are congruent if you can slide them around, rotate them, and flip them over in a lot of ways so they make a pile where they exactly fit over each other.
AAA does not work.
If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not essentially the same size. They are in this particular case known as similar triangles.
SSA does not work.
If two sides of a triangle are given and a non-included angle is also given, it is possible to draw two different triangles that satisfy the values. It is thus not enough to prove congruence.
Constructions
Another way to think about the above is to find out if it is possible to construct a unique triangle with the information you are provided with. For instance, If you were given the lengths of two sides and the included angle (SAS), there is only one possible triangle you could draw. If you drew two of them, they would be the same shape and size – the definition of congruent.
Properties of Congruent Triangles
If two triangles are congruent, tin that case each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.
To remember this significant idea, a few people find it of assistance to make use of the acronym CPCTC, which stands for “Corresponding Parts of Congruent Triangles are Congruent”.
In addition to sides and angles, all other properties of the triangle are the same as well like the area, perimeter, location of centers, circles and so on etc.
Congruent Triangles – Why AAA doesn’t work
Having all three corresponding angles equal is not enough to prove congruence. It entails that the fact that two triangles have congruent corresponding angles does not prove the triangles are congruent.
But they are similar
Triangles like this that are the same shape but different sizes are referred to as similar triangles.
Congruent Triangles – Why SSA doesn’t work
Given two sides and non-included angle (SSA) is not enough to prove congruence.
You may be tempted to think that given two sides and a non-included angle is enough to prove congruence. But there are two triangles possible that have the same values, therefore, SSA is not sufficient to prove congruence.
In the figure above, the two triangles above are initially congruent. But if you “Show other triangle” you will see that there is another triangle that is not congruent but that still satisfies the SSA condition. AB is the same length as PQ, BC is the same length as QR, and the angle A is the same measure as P. And yet the triangles are clearly not congruent – they have a different shape and size.
Does that mean that I can’t use SSA at all?
You can never use SSA on its own. But you could use it if you also provide proof as to which of the two possible triangles are described.
Similar Triangles
Triangles are similar if they have the same shape, but can be different sizes. They are still similar even if one is rotated, or one is a mirror image of the other as shown below.
In this specific example, the triangles are the same size, for this reason, they are as well congruent.
Parallelograms on Same Base and between Same Parallels
Assuming that in the parallelogram figure,
ABCD and BCEF are the two parallelograms on the same base BC and between the parallels BC and AE.
Thus, area of parallelogram ABCD = Area of parallelogram BCEF.
Explanation:
Draw a parallelogram ABCD on a thick sheet of paper or a cardboard sheet. Now, draw a line segment DE by linking the point D and E.
Next, cut a triangle A’D’E’ congruent to triangle ADE in a different sheet with the assistance of a tracing paper and place ∆ A’D’E’ in such a way that A’D’ coincides with BC as illustrated.
Note: There are two parallelograms ABCD and EE’CD on the same base DC and between the same parallels AE’ and DC. What can you say about their areas?
Just like ∆ ADE ≅ ∆ A’ D’ E’
Thus Area (ADE) = Area (A’ D’ E’)
Again Area (ABCD) = Area (ADE) + Area (EBCD)
= Area (A’D’E’) + Area (EBCD)
= Area (EE’CD)
Therefore, the two parallelograms are equal in area.
Solved Example:
Parallelograms ABCD and ABEF are situated on the opposite sides of AB in such a way that D, A, F are not collinear. Prove that DCEF is a parallelogram, and parallelogram ABCD + parallelogram ABEF = parallelogram DCEF.
Construction: D, F and C, E are joined.
Proof: AB and DC are two opposite sides of parallelogram ABCD,
Therefore, AB ∥ DC and AB = DC
Again, AB and EF are two opposite sides of parallelogram ABEF
Therefore, AB ∥ EF and AB ∥ EF
Therefore, DC ∥ EF and DC = EF
Therefore, DCEF is a parallelogram.
Therefore, ∆ ADF and ∆ BCE, we OBTAIN
AD = BC (opposite sides of parallelogram ABCD)
AF = BE (opposite sides of parallelogram ABEF)
And DF = CE (opposite sides of parallelogram CDEF)
Therefore, ∆ ADF ≅ ∆ BCE (side – side – side)
Therefore, ∆ ADF = ∆ BCE
Thus, polygon AFECD – ∆BCE = polygon AFCED – ∆ ADF
Parallelogram ABCD + Parallelogram ABEF = Parallelogram DCEF
Sum of Interior Angles of a Polygon Sum of Interior Angles of a Polygon== 180(n – 2)(where n = number of sides)
There are two types of questions that come up when making use of this formula:
1. Questions that ask you to find the number of degrees in the sum of the interior angles of a polygon.
2. Questions that ask you to find the number of sides of a polygon.
Clue: When working with the angle formulas for polygons, be sure to read each question carefully for clues as to which formula you will be required to use to solve the problem. Look for the words that explain each kind of formula, like the words sum, interior, each, exterior and degrees.
Example 1: Find the number of degrees in the sum of the interior angles of an octagon.
An octagon has 8 sides. So n = 8. Using the formula from above, 180(n – 2) = 180 (8 – 2) = 180 (6) = 1080 degrees.
Example 2: How many sides does a polygon have if the sum of its interior angles is 720°?
Since, the number of degrees is given, set the formula above equal to 720°, and solve for n.
180 (n – 2) = 720 n – 2 = 4 n = 6 Set the formula = 720° Divide both sides by 180 Add 2 to both sides
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