what theorem can abdul use to determine the two triangles are similar? (6 points)
Similar triangles are triangles that have the same shape, just their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion. We denote the similarity of triangles here past '~' symbol.
Definition
Two triangles are similar if they accept the same ratio of corresponding sides and equal pair of corresponding angles.
If two or more than figures have the same shape, but their sizes are different, then such objects are called similar figures. Consider a hula hoop and bicycle of a cycle, the shapes of both these objects are similar to each other as their shapes are the same.
In the figure given in a higher place, two circles C1 and Cii with radius R and r respectively are similar every bit they have the aforementioned shape, but necessarily non the same size. Thus, we tin can say that C1~ C2.
It is to be noted that, two circles always take the aforementioned shape, irrespective of their bore. Thus, two circles are ever like.
Triangle is the three-sided polygon. The condition for the similarity of triangles is;
i) Corresponding angles of both the triangles are equal, and
ii) Corresponding sides of both the triangles are in proportion to each other.
Similar Triangle Case
In the given figure, two triangles ΔABC and ΔXYZ are similar but if,
i) ∠A = ∠10, ∠B = ∠Y and ∠C = ∠Z
ii) AB/XY = BC/YZ = AC/XZ(Similar triangles proportions)
Hence, if the above-mentioned weather condition are satisfied, and so we tin say that ΔABC ~ ΔXYZ
It is interesting to know that if the respective angles of two triangles are equal, then such triangles are known as equiangular triangles. For two equiangular triangles we tin country the Basic Proportionality Theorem (better known as Thales Theorem) as follows:
- For ii equiangular triangles, the ratio of any two corresponding sides is always the same.
Properties
- Both accept the same shape but sizes may be different
- Each pair of corresponding angles are equal
- The ratio of corresponding sides is the same
Formulas
According to the definition, two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional. Hence, we tin can detect the dimensions of one triangle with the help of some other triangle. If ABC and XYZ are 2 similar triangles, and then by the aid of below-given formulas, we can detect the relevant angles and side lengths.
- ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
- AB/XY = BC/YZ = AC/XZ
Once we have known all the dimensions and angles of triangles, it is easy to observe the area of like triangles.
Similar Triangles and Congruent Triangles
The comparing of similar triangles and congruent triangles is given below in the table.
| Like Triangles | Congruent Triangles |
| They are the aforementioned shape just unlike in size | They are the same in shape and size |
| Symbol is '~' | Symbol is ' ≅' |
| Ratio of all the corresponding sides are same | Ratio of corresponding sides are equal to a constant value |
To Know how to Find the Area Of Similar Triangles, Watch The Below Video:
Similar triangles Theorems with Proofs
Allow us acquire here the theorems used to solve the issues based on similar triangles along with the proofs for each.
AA (or AAA) or Angle-Angle Similarity
If any two angles of a triangle are equal to any 2 angles of some other triangle, so the two triangles are like to each other.
From the figure given above, if ∠ A = ∠X and ∠C = ∠Z so ΔABC ~ΔXYZ.
From the result obtained, we can hands say that,
AB/XY = BC/YZ = Air conditioning/XZ
and ∠B = ∠Y
SAS or Side-Angle-Side Similarity
If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, and then two triangles are said to be like.
Thus, if ∠A = ∠X and AB/XY = Air-conditioning/XZand then ΔABC ~ΔXYZ.
From the congruency,
AB/XY = BC/YZ = Air conditioning/XZ
and ∠B = ∠Y and ∠C = ∠Z
SSS or Side-Side-Side Similarity
If all the three sides of a triangle are in proportion to the three sides of another triangle, and so the ii triangles are like.
Thus, if AB/XY = BC/YZ = AC/XZ so ΔABC ~ΔXYZ.
From this outcome, we tin infer that-
∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
Likewise, read:
- Isosceles Triangle Equilateral
- Area Of A Triangle
- Congruence Of Triangles Class 9
- Important Questions Class 10 Maths Affiliate six Triangles
Trouble and Solutions
Let us get through an example to understand it better.
Q.1: In theΔABC length of the sides are given as AP = 5 cm , Pb = 10 cm and BC = 20 cm. As well PQ||BC. Observe PQ.
Solution: In ΔABC and ΔAPQ, ∠PAQ is common and ∠APQ = ∠ABC (corresponding angles)
⇒ ΔABC ~ ΔAPQ (AA criterion for similar triangles)
⇒ AP/AB =PQ/BC
⇒ five/15 = PQ/20
⇒ PQ = 20/3 cm
Q.2: Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the signal O. Using a similarity benchmark for two triangles, evidence that AO/OC = OB/OD.
Solution: ABCD is a trapezium and O is the intersection of diagonals AC and BD.
In ΔDOC and ΔBOA,
AB || CD, thus alternating interior angles will be equal,
∴∠CDO = ∠ABO
Similarly,
∠DCO = ∠BAO
Also, for the two triangles ΔDOC and ΔBOA, vertically opposite angles will be equal;
∴∠Doc = ∠BOA
Hence, by AAA similarity criterion,
ΔDOC ~ ΔBOA
Thus, the corresponding sides are proportional.
DO/BO = OC/OA
⇒OA/OC = OB/OD
Hence, proved.
Q.iii: Check if the two triangles are like.
Solution: In triangle PQR, by bending sum belongings;
∠P + ∠Q + ∠R = 180°
lx° + 70° + ∠R = 180°
130° + ∠R = 180°
Decrease both sides by 130°.
∠ R= 50°
Once more in triangle XYZ, past angle sum property;
∠Ten + ∠Y + ∠Z = 180°
∠60° + ∠Y + ∠50°= 180°
∠ 110° + ∠Y = 180 °
Decrease both sides by 110°
∠ Y = 70°
Since,∠Q = ∠ Y = 70° and ∠Z = ∠ R= 50°
Therefore, past Angle-Angle (AA) rule,
ΔPQR~ΔXYZ.
Similar Triangles Video Lesson
This video will aid you visualize basic criteria for the similarity of triangles.
To larn more than about similar triangles and properties of similar triangles, download BYJU'South- The Learning App.
Oftentimes Asked Questions – FAQs
What are like triangles?
Two triangles are like if they take the aforementioned ratio of corresponding sides and equal pair of corresponding angles.
What is the symbol for similar triangles?
If ABC and PQR are two like triangles, then they are represented by:
∆ABC ~ ∆PQR
Are like triangles and congruent triangles same?
Similar triangles take the same shape but sizes may vary but congruent triangles have the same shape and size. Coinciding triangles are represented past symbol '≅'.
What are three similarities theorems for triangles?
The three similarities theorem are:
Angle-bending (AA)
Side-bending-side (SAS)
Side-side-side (SSS)
How to find the proportion of similar triangles?
If two triangles are similar and have sides A,B,C and a,b,c, respectively, and then the pair of corresponding sides are proportional, i.e.,
A : a = B : b = C : c.
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