Question

Theorem: If the acute angles of a right angled triangle have measures 30° and 60°, then
the length of the side opposite to 30° angle is half the length of the hypotenuse,
Fill in the blanks and complete the proof)
Given : In A ABC
ZB - 90°. ZC-30°, ZA= 60°
B
C
60
B
C С
D
To prove : AB АС
Fig. 3.29
Construction : Take a point D on the extended
seg AB such that AB = BD. Draw seg DC.
Proof : A ABC and A DBC
seg AB seg DB .........
ZABC - ZDBC.......
seg BC seg BC ...........
..A ABC = A DBC
Fig. 3.30
.. ZBAC ZBDC ........ (c.ac.t.)
In A ABC, ZBAC = 60°. ZBDC = 60°
ZDAC = ZADC = ZACD = 60°..., sum of angles of A ADC is 180°
HA ADC is an equilateral triangle.
.. AC = AD-DC ........ corollary of converse of isosceles triangle theorem
But AB AD...... construction
2.
..АВ AC ..... AD - AC
2
12 늘
1
Activity
With the help of the Figure 3.29 above fill in the blanks and complete the proof of th
following theorem
Theorem: If the acute angles of a right angled triangle have measures 30°and 60° the
V3
the length of the side opposite to 60° angle is
2
X hypotenuse
Proof: In the above theorem we have proved AB AC
AB+ BC =
........ Pythagoras theorem
AC + BC
..BCP-AC-AC
BC -
2
.: BC
V3
2
AC​

Answer 1

Answer:

Construction : Take a point D on the extended

seg AB such that AB = BD. Draw seg DC.

Proof : A ABC and A DBC

seg AB seg DB .........

ZABC - ZDBC.......

seg BC seg BC ...........

..A ABC = A DBC

Fig. 3.30

.. ZBAC ZBDC ........ (c.ac.t.)

In A ABC, ZBAC = 60°. ZBDC = 60°

ZDAC = ZADC = ZACD = 60°..., sum of angles of A ADC is 180°

HA ADC is an equilateral triangle.

.. AC = AD-DC ........ corollary of converse of isosceles triangle theorem

But AB AD...... construction

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