Triangles online mcq quiz
Triangles boostup points
➥ Similar and Congruent Figures
o Two geometric figures having the same shape and size are said to be congruent figures.
o Two geometric figures having the same shape, but not necessarily the same size, are called similar figures.
Example:
(1) All circles are similar.
(2) All equilateral triangles are similar.
(3) All congruent figures are similar. However, the converse is not true.
➥ Similarity of Polygons
Two polygons with the same number of sides are similar, if
o their corresponding angles are equal
o their corresponding sides are in the same ratio (or proportion)
➥ Two lines segments are congruent, if they are equal in length.
➥ Two angles are congruent, if they have the same measure.
➥ CPCT: CPCT stands for Corresponding Parts of Congruent Triangles. If ΔABC ΔPQR, then corresponding sides are equal i.e., AB = PQ, BC = QR, and CA = RP and corresponding angles are equal i.e., ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.
➥ Basic proportionality theorem:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
➥Converse of basic proportionality theorem:
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
➥ Theorem: (AAA similarity criterion)
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence, the two triangles are similar.
➥ Theorem: (AA similarity criterion)
If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.
➥ Theorem: (SSS similarity criterion)
If in two triangles, sides of one triangle are proportional to the sides of the other triangle then the two triangles are similar by SSS similarity criterion.
➥ Theorem: (SAS similarity criterion)
If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar by SAS similarity criterion.
➥ Areas of similar triangles
Theorem: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Result: Using the above theorem, the result “ the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians or altitudes or angle bisector” can be proved.
➥ Pythagoras theorem: 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:
In a triangle, if the square of one side is equal to the sum of the squares of other two sides, then the angle opposite to the first side is a right angle.
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