When two triangles are the corresponding angles have the same measure and the corresponding sides are proportional?

Corresponding sides of a polygon are the sides that are in the same position in similar polygons. In geometry, finding the congruence and similarity involves comparing corresponding sides and corresponding angles of the polygons. In this article, let's learn more about similar right triangles, corresponding sides, their definition, how they are proportional, the differences between congruent and similar triangles with a few solved examples. 

Corresponding sides are the sides that are in the same position in any different 2-dimensional shapes. For any two polygons to be congruent, they must have exactly the same shape and size. This means that all their interior angles and their corresponding sides must be the same measure. For any two polygons to be similar, the ratios of the lengths of each pair of corresponding sides must be equal. Let us consider 2 quadrilaterals ABCD and PQRS to understand the corresponding sides.

From the above image, we can observe that:

• The side AB corresponds to the side PQ
• The side BC corresponds to the side QR
• The side CD corresponds to the side RS
• The side DA corresponds to the side SP

The SSS - Congruence rule states that, in two triangles, if all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles can be considered to be congruent. In a triangle, the corresponding sides are the sides that are in the same position in different triangles. In the below-given images, the two triangles are congruent and their corresponding sides are color-coded.

In the above two triangles ABC and XYZ,

• AB is the corresponding side to XY
• BC is the corresponding side to YZ
• CA is the corresponding side to ZX

The congruent triangles are different from similar triangles considering the aspect of corresponding sides. The table below shows the differences between congruent and similar triangles with the help of the illustration.

If the two shapes are similar, then their corresponding sides are proportional. In two similar triangles, the corresponding sides are proportional and these corresponding sides always touch the same two angle pairs. In the given similar triangles PQR and STU:

• PQ is the corresponding side to ST, and while PQ touches ∠P and ∠Q, ST touches ∠S and ∠T
• PR is the corresponding side to SU, and while PR touches ∠P and ∠R, SU touches ∠S and ∠U
• QR is the corresponding side to TU, and while QR touches ∠Q and ∠R, TU touches ∠T and ∠U

To understand proportionality, consider a)  \(\triangle \text{ABC} \simeq \triangle \text{ADE}\)

AB/AD = AC/AE

AB × AE = AD × AC

Consider b) \(\triangle \text{PQR} \simeq \triangle \text{STU}\)

PQ/ST = PR/SU = QR/TU

Hence, if two triangles are similar, then their corresponding sides are proportional.

Consider two similar triangles ABC and DEF,

In the above image,

AB/DE = BC/EF

10/16 = 9/a

10 × a = 16 × 9

a = (16 × 9)/10

a = 144/10 = 14.4

Thus we conclude that if \(\triangle \text{ABC} \simeq \text{DEF}\), then we say that the corresponding sides are proportional and the angles are equal.

AB/DE = BC/EF = CA/FD = k, where k is the equivalent ratio or the trigonometric ratio.

If the lengths of the hypotenuse and a leg of one right-angled triangle are proportional to the corresponding parts of the other right triangle, then the triangles are similar. Consider the two right triangles ABC and DEF in the below-given image,

\[\dfrac{\text{The shortest side of the small triangle}}{\text{The shortest side of the large triangle}}\\=\dfrac {\text{The longest side of the small triangle}} {\text{The longest side of the large triangle}}\\= \dfrac{\text{Hypotenuse of small triangle}}{\text{Hypotenuse of the large triangle}}\]

a/d = b/e = c/f

Related Articles on Corresponding Sides

Check out the following pages related to the corresponding sides.

• Congruence in Triangles
• Isosceles Triangles
• Perimeter of isosceles triangle

Important Notes

Here is a list of a few points that should be remembered while studying corresponding sides:

• When two triangles are similar, the ratios of the lengths of their corresponding sides are equal.
• Two triangles are considered to be congruent if all their corresponding angles and sides are equal
• The SSS - Congruence rule states that, in two triangles, if all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles can be considered to be congruent.

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Consider this example: if one polygon has sequential sides p,q, and r, and the other has sequential sides a,b, and c, and if q and b are corresponding sides, then side p (adjacent to q) must correspond to either a or c (both adjacent to b).

Define Corresponding Sides and Angles.

Sides and angles can be considered as corresponding when a pair of matching angles or sides are in the same position in two different shapes.

What Are the Corresponding Parts of Congruent Triangles?

In two congruent triangles, the sides and angles are considered to be their corresponding parts. The corresponding parts are found in the same relative positions.

What Is the Difference Between Corresponding and Alternate Angles?

Comparing the two angles in 2 similar polygons, the corresponding angles relatively occupy the same position. When a transversal meets two parallel lines, corresponding angles that lie relatively in the same position are considered to be congruent, they are of the same measure. Angles are considered to be alternate angles when they are on the opposite sides of the transversal lines.

What Letter of the Alphabet Has Corresponding Angles?

The letter F is identified to get corresponding angles. The corresponding angles are relatively in the same position when a transversal intersects two parallel lines and they are equal.

There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.

AA (Angle-Angle)

If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear.

Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.

SAS (Side-Angle-Side)

If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional.

Conclusion

These are the main techniques for proving congruence and similarity. With these tools, we can now do two things.

• Given limited information about two geometric figures, we may be able to prove their congruence or similarity.
• Given that figures are congruent or similar, we can deduce information about their corresponding parts that we didn't previously know.

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