Special Right Triangles
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Key Questions

There are 2 types of special right triangles.
Type 1. Triangle that is half of a equilateral triangle. Its 3 angle measures are: 30, 60 and 90 deg. Its side measures are : a, a/2; and (a*sqr.3)/2.
Type 2. Triangle that has its side measures in the ratio of 3:4:5. The proof is given by the Pythagor theorem: c^2 = b^2 + a^2.
Use of special right triangles.
In the old time, people use the special right triangles with sides ratio 3:4:5 to figure out, in the field, a right angle or a rectangular, or square, shape.
Now, students just use the properties of special right triangle to find, by computing, the unknown sides or angles. 
Answer:
Consider the properties of the sides, the angles and the symmetry.
Explanation:
#454590" "# refers to the angles of the triangle.The
#color(blue)("sum of the angles is " 180Â°)# There are
#color(blue)("two equal angles")# , so this is an isosceles triangle.It therefore also has
#color(blue)(" two equal sides.")# The third angle is
#90Â°# . It is a#color(blue)("rightangled triangle")# therefore Pythagoras' Theorem can be used.The
#color(blue)("sides are in the ratio " 1 :1: sqrt2)# It has
#color(blue)("one line of symmetry")#  the perpendicular bisector of the base (the hypotenuse) passes through the vertex, (the#90Â°# angle).It has
#color(blue)("no rotational symmetry.")# 
#mathbf{30^circ""60^circ""90^circ}# TriangleThe ratios of three sides of a
#30^circ""60^circ""90^circ# triangle are:#1:sqrt{3}:2#
I hope that this was helpful.

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