Triangles is a closed figure with three straight sides and three angles.
Properties of Triangle
Congruent Triangles
Those triangles which have exactly the same three sides and exactly the same three angles. If the figures have exactly the same shape and size are called congruent figures.
Conditions of congruency of triangles
Two triangles will be congruent under the following conditions
Similar Triangles
The figures which are exactly the same in the shape but sizes may be different are known as similar figures. In the given figure,\(\triangle\)ABC and\(\triangle\)PQR are similar triangles because they have equal angles and the same shape.
Thus, if three angles of one triangle are respectively equal to three angles of another triangle the are said to be similar.\(\triangle\)ABC is similar to\(\triangle\)PQR is written as\(\triangle\)ABC∼\(\triangle\)PQR. The symbol '∼' is used to denote is similar to.
Here AB and PQ, BC and QR, CA and RP are the corresponding sides of the similar triangle.
The corresponding sides of similar triangles are always proportional, i.e the ratios of the corresponding sides are equal.
\(\therefore\) \(\frac{AB}{PQ}\) =\(\frac{BC}{QR}\) =\(\frac{CA}{RP}\)
Properties of Triangle
Solution:
Here, x° + 2x° + 90° = 180° [Being the sum of the angles of right-angled triangle.]
or, 3x° = 180° - 90°
or, x° = \(\frac{90°}{3}\) = 30°
\(\therefore\) x° = 30° and 2x° = 2×30° = 60°
Solution:
Let the angles of the triangle are 3x°, 4x° and 5x° respectively.
Now, 3x° + 4x° + 5x° = 180° [Being the sum of the angles of a triangle.]
or, 12x° = 180°
or, x° = \(\frac{180°}{12}\) = 15°
\(\therefore\) 3x° = 3×15 = 45°, 4x° = 4×15° =60° and 5x° = 5×15° = 75°.
Solution:
x+108° = 180° [Being the sum a straight angle]
or, x = 180° - 108° = 72°
y = x = 72° [Being the base angles of an isoceles triangle]
Again, y + z = 108° [Being the sum equal to exterior angle of the triangle]
or, 72°+z = 108°
or, z = 108° - 72° = 36°
\(\therefore\) x= y = 72° and z = 36°.
Solution:
In \(\triangle\)^{s}ABC and PQR,
Solution:
Here, \(\triangle\)DEF ∼ \(\triangle\)XYZ
\(\therefore\) \(\frac{DE}{XY}\) = \(\frac{EF}{YZ}\)
or, \(\frac{6}{4}\) = \(\frac{3}{YZ}\)
or, 6YZ = 12
or, YZ = \(\frac{12}{6}\) = 2 cm
If all sides are equal in a triangle then it is known as _____ triangle.
The triangle having any two sides are equal is known as ______ triangle.
If the triangle having different-different sides is known as ______ triangle.
If x° and 2x° are two acute angles of right-angled triangle find the value of x.
ASK ANY QUESTION ON Geometry Triangles
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