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Short Notes: Triangles
Similar Figures
Two figures having the same shape but not necessarily the same size are called similar figures.- Corresponding angles of similar figures are equal.
- Corresponding lengths of similar figures are proportional.
- All congruent figures are similar, but not all similar figures are congruent.
For Example:
(i) Any two line segments are similar since length are proportional
(ii) Any two circles are similar since radii are proportional
(iii) Any two squares are similar since corresponding angles are equal and lengths are proportional.
Note: Similar figures are congruent if there is one to one correspondence between the figures that preserves both shape and size.
Try yourself: Which of the following pairs of figures are similar?MULTIPLE CHOICE QUESTION
Similar Polygons
Two polygons are said to be similar to each other, if:- Their corresponding angles are equal, and
- The lengths of their corresponding sides are proportional
For Example:
From above we deduce:
Two triangles are similar, if their
(i) Corresponding angles are equal
- ∠A = ∠P
- ∠B = ∠Q
- ∠C = ∠R
(ii) Corresponding sides are proportional
Thales' Theorem or Basic Proportionality Theorem
Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Proof of Thales Theorem
Given: In ∆ABC, DE || BC.
To prove: AD/DB = AE/EC
Const.: Draw EM ⊥ AD and DN ⊥ AE. Join B to E and C to D.
Proof: In ∆ADE and ∆BDE,
........(i) [Area of ∆ = 1/2 x base x corresponding altitude]
In ∆ADE and ∆CDE,
∵ DE || BC ...[Given ]
∴ ar(∆BDE) = ar(∆CDE) ...[∵ As on the same base and between the same parallel sides are equal in area]
From (i), (ii) and (iii),
Hence proved.
Try yourself: Two triangles have the same shape, and their corresponding angles are equal. If one triangle has a side length of 6 centimeters and the other triangle has a side length of 9 centimeters, can we conclude that these triangles are similar? Why or why not?MULTIPLE CHOICE QUESTION
Criterion for Similarity of Triangles
Two triangles are similar if either of the following three criteria are satisfied:
- AAA similarity Criterion: If two triangles are equiangular, then they are similar.
- Corollary (AA similarity): If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
- SSS Similarity Criterion: If the corresponding sides of two triangles are proportional, then they are similar.
- SAS Similarity Criterion: If in two triangles, one pair of corresponding sides are proportional and the included angles are equal, then the two triangles are similar.
Results in Similar Triangles based on Similarity Criterion:
- Ratio of corresponding sides = Ratio of corresponding perimeters
- Ratio of corresponding sides = Ratio of corresponding medians
- Ratio of corresponding sides = Ratio of corresponding altitudes
- Ratio of corresponding sides = Ratio of corresponding angle bisector segments
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FAQs on Short Notes: Triangles
| 1. What are similar figures and how can we identify them? |  |
Ans.Similar figures are shapes that have the same shape but may differ in size. To identify similar figures, we compare their corresponding angles and sides. If the corresponding angles are equal and the lengths of corresponding sides are in proportion, then the figures are similar.
| 2. What defines similar polygons and how do we determine if two polygons are similar? | |
Ans.Similar polygons are polygons that have equal corresponding angles and proportional corresponding side lengths. To determine if two polygons are similar, we can either compare their angles and sides directly or use the ratio of the lengths of corresponding sides to check for proportionality.
| 3. What is Thales' Theorem and how does it relate to similar triangles? | |
Ans.Thales' Theorem states that if A, B, and C are points on a circle where the line segment AC is the diameter, then the angle ABC is a right angle. This theorem implies that triangles formed under this condition are similar to other right triangles, providing a basis for comparing and establishing the proportionality of sides.
| 4. What is the criterion for the similarity of triangles? | |
Ans.The criteria for the similarity of triangles include the Angle-Angle (AA) criterion, where two triangles are similar if two angles of one triangle are equal to two angles of another triangle. The Side-Angle-Side (SAS) criterion states that if an angle of one triangle is equal to an angle of another triangle and the sides including these angles are in proportion, then the triangles are similar.
| 5. How can we apply the Basic Proportionality Theorem in real-life situations? | |
Ans.The Basic Proportionality Theorem, also known as Thales' theorem, can be applied in various real-life situations such as in architecture, where it helps in creating scale models, or in navigation, where it assists in determining distances and angles. By ensuring that segments are proportional, we can maintain the correct proportions in designs and measurements.
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Document Description: Short Notes: Triangles for Class 10 2026 is part of Mathematics (Maths) Class 10 preparation. The notes and questions for Short Notes: Triangles have been prepared according to the Class 10 exam syllabus. Information about Short Notes: Triangles covers topics like and Short Notes: Triangles Example, for Class 10 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Short Notes: Triangles.
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