Question

Similarities of triangle ......
Notes

Chota hi ho prr smjhne layak dena ....​

Answer 1

Step-by-step explanation:

Two triangles are said to be similar when they have two corresponding angles congruent and the sides proportional. ... The area, altitude, and volume of Similar triangles are in the same ratio as the ratio of the length of their sides.

Answer 2

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SIMILAR FIGURES

• Two figures having the same shape but not necessary the same size are called similar figures.
• All congruent figures are similar but all similar figures are not congruent.

SIMILAR POLYGONS

Two polygons are said to be similar to each other, if:

• (i) their corresponding angles are equal, and
• (ii) the lengths of their corresponding sides are proportional

THALES THEOREM OR BASIC PROPORTIONALITY THEORY

Theorem 1:

Statement:

• 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.

CRITERION FOR SIMILARITY OF TRIANGLES

• Two triangles are similar if either of the following three criterion’s 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.

AREA OF SIMILAR TRIANGLES

Theorem 2.

• The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Results based on Area Theorem:

• Ratio of areas of two similar triangles = Ratio of squares of corresponding altitudes
• Ratio of areas of two similar triangles = Ratio of squares of corresponding medians
• Ratio of areas of two similar triangles = Ratio of squares of corresponding angle bisector segments.

Note:

• If the areas of two similar triangles are equal, the triangles are congruent.

PYTHAGORAS THEOREM

Theorem 3:

Statement:

• Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

CONVERSE OF PYTHAGORAS THEOREM

Theorem 4:

Statement:

• Prove that, in a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.