In geometry, finding the congruence and similarity involves comparing corresponding sides and corresponding angles of the polygons.
Corresponding sides are the sides that are in the same position in different 2-dimensional shapes.
Consider the 2 quadrilaterals ABCD and PQRS in the image below.
AB corresponds to PQ, BC corresponds to QR, CD corresponds to RS and DA corresponds to SP.
In this mini-lesson, we will learn about the corresponding sides definition, similar triangles, similar right triangles, and congruent by knowing the interesting facts around them.
Lesson Plan
What are Corresponding Sides of Triangles?
Corresponding Sides Definition
Corresponding sides of a triangle are the sides that are in the same position in different triangles.
Consider two triangles ABC and LMN in the image below. Start with any side, say AB in the \(\triangle ABC\).
Locate the similar side corresponding to this AB.
We find with respect to its position, that LM is the corresponding side to AB in \(\triangle LMN\).
Likewise, we identify that BC in \(\triangle ABC\) is the corresponding side to MN in \(\triangle LMN\) and
CA is the corresponding side to NL.
Thus locating one side corresponding to the other helps us to identify the other corresponding sides of the two triangles.
Congruent Triangles vs Similar Triangles
The congruent triangles are different from similar triangles considering the aspect of corresponding sides.
Congruent Triangles | Similar Triangles |
---|---|
Two triangles are congruent if they have all their corresponding angles and sides equal. | Two triangles are similar if they have all their corresponding angles equal and their corresponding sides are in the same ratio. |
In \(\triangle\)ABC and \(\triangle\)LMN , (1) AB = LM, BC = MN, and AC = LN. (2) \(\angle\) A = \(\angle\) M, \(\angle\) B = \(\angle\) L, \(\angle\) C = \(\angle\)N \(\therefore \triangle\) ABC \(\cong \triangle\) LMN |
In \(\triangle\)PQR and \(\triangle\)SUT, (1)PQ \(\propto\) ST, QR \(\propto\) TU, and PR \(\propto\) SU (2) \(\angle\) P = \(\angle\) S, \(\angle\) Q = \(\angle\) T, \(\angle\) R = \(\angle\) U \(\therefore \triangle\) PQR \(\simeq \triangle\) STU |
How Do You Know If The Corresponding Sides Are Proportional?
If the two shapes are similar, then their corresponding sides are proportional.
Similar Triangles
If an angle of one triangle is equal to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.
Consider a) \(\triangle \text{ABC} \simeq \triangle \text{ADE}\)
\[\dfrac{\text{AB}}{\text{AD}} = \dfrac{\text{AC}}{\text{AE}}\]
\[\text{AB}\times \text{AE} = \times{AD}\times \text{AC}\]
Consider b) \(\triangle \text{PQR} \simeq \triangle \text{STU}\)
\[\dfrac{\text{PQ}}{\text{ST}} = \dfrac{\text{PR}}{\text{SU}} =\dfrac{\text{QR}}{\text{TU}}\]
Consider triangles ABC and DEF.
If two triangles are similar, then their corresponding sides are proportional.
\[\begin{align}\dfrac{\text{AB}}{\text{DE}} &= \dfrac{\text{BC}}{\text{EF}}\\\\\dfrac{10}{16} &= \dfrac{9}{a}\\\\ 10 \times a &= 16 \times 9\\\\ a&= \dfrac{(16\times 9)}{10}\\\\ &=\dfrac{144}{10}\\\\ &= 14.4\end{align}\]
In the following simulation, the two triangles ABC and PQR are similar.
Drag the points A, B, C.
What can you notice about the triangles?
Verify for yourself by sliding the points that the equivalent sides are always proportional.
Thus we conclude that if \(\triangle \text{ABC} \simeq \text{PQR}\), then we say that the corresponding sides are proportional and the angles are equal.
\(\dfrac{\text{AB}}{\text{PQ}} = \dfrac{\text{BC}}{\text{QR}} =\dfrac{\text{CA}}{\text{RP}} = k\) |
where k is the equivalent ratio or the trigonometric ratio.
- Locate one corresponding side and the others will fall in sequence.
- Finding the corresponding sides helps us find the corresponding angles as well, as the angle is subtended between two sides.
Corresponding Sides in Right Triangles
If the lengths of the hypotenuse and a leg of one right triangle are proportional to the corresponding parts of the other right triangle, then the triangles are similar.
\[\dfrac{\text{The shortest side of the small triangle}}{\text{The shortest side of the large triangle}}\\=\dfrac {\text{The longest side of the small triangle}} {\text{The longest side of the large triangle}}\\= \dfrac{\text{Hypotenuse of small triangle}}{\text{Hypotenuse of the large triangle}}\]
\[\dfrac{\text{a}}{\text{d}}=\dfrac {\text{b}} {\text{e}}= \dfrac{\text{c}}{\text{f}}\]
- When two triangles are similar, the ratios of the lengths of their corresponding sides are equal.
- To determine if the triangles shown are similar, compare their corresponding sides.
- If two triangles have three pairs of sides in the same ratio, then the triangles are similar by SSS property.
Solved Examples
Example 1 |
Brandon wants to check if the two triangles are similar. Can you help him?
Solution
To determine if the triangles are similar we need to check if the sides are proportional.
Identify the corresponding sides in \(\triangle\)LMN and \(\triangle\)XYZ.
\(\dfrac{LN}{XY} = \dfrac{30}{40} = \dfrac{30\div 10}{40\div 10} = \dfrac{3}{4}\)
\(\dfrac{LM}{YZ} = \dfrac{42}{56}= \dfrac{42\div 14}{56\div 14} = \dfrac{3}{4}\)
\(\dfrac{MN}{ZX} = \dfrac{54}{72} = \dfrac{54\div 9}{72\div 9} = \dfrac{3}{4}\)
We find that the corresponding sides are proportional to each other.
\(\therefore\) The two triangles are similar. |
Example 2 |
Ria was arranging a few 2-dimensional shapes while she was constructing a math puzzle. She found 2 triangles arranged in the way shown below. On observation, she found that the triangles are similar. She wanted to know which are the corresponding sides of these triangles. How can you help her?
Solution
AOB and POQ are the two triangles.
Known that they are similar.
If the triangles are similar, the sides are proportional.
Check which two sides form the equal proportion.
We find that \(\dfrac{3}{6}=\dfrac{1}{2}\\\dfrac{4}{8}=\dfrac{1}{2}\\\dfrac{5}{10}=\dfrac{1}{2}\)
\(\dfrac{OA}{OP}=\dfrac{OB}{OQ}=\dfrac{AB}{PQ}=\dfrac{1}{2}\)
\(\therefore\) OA corresponds to OP, OB corresponds to OQ and AB corresponds to PQ. |
Example 3 |
Marc has made a paper fold for his project. He wants to know the side 'f' if \(\triangle\) OPN and \(\triangle\) OQM are similar. How can you help him?
Solution
OP = 4 inches
OQ = 8 inches
Since the two triangles are similar, the ratios of the lengths of their corresponding sides are equal.
\[\begin{align}\dfrac{\text{4}}{\text{8}}&=\dfrac {\text{3}} {\text{f}}\\ 4 \times\text{f} &= 3\times 8 \\\text{f}&= \dfrac{24}{4}\\ \text{f} &= 6 \text{ inches}\end{align}\]
\(\therefore\) the unknown side \(\text{f} = 6 \text{ inches}\) |
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The mini-lesson targeted the fascinating concept of corresponding sides. The math journey around corresponding sides starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
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Frequently Asked Questions (FAQs)
1. What are similar triangles?
Two triangles are said to be similar if their corresponding angles are equal and the corresponding sides are proportional.
2. Are corresponding sides equal?
Two congruent triangles have their corresponding sides equal. Two similar triangles have their corresponding sides proportional.