Page 27 - Mathematics_Form_Two

Congruence

Proof But ABC + CBD 180= °
ˆ
ˆ
is common to both sides of the
Construction: Draw line XY through C
parallel to AB. Thus, equation. ˆ ˆ ˆ
ˆ
ˆ
ACX = CAB (i) Thus, CAB BCA+ = CBD .
(alternate interior angles as XY / /AB ) Therefore, the sum of two interior angles
of a triangle is equal to the exterior angle
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ˆ
ˆ
YCB = ABC (ii) of the third interior angle.
(alternate interior angles as XY / /AB )
But ACX BCA YCB 180+ ˆ ˆ + ˆ = ° (iii) Example 2�3
(degree measure of a straight angle) Prove that the bisectors of the angles Mathematics for Secondary Schools
Substitute (i) and (ii) into (iii) to get formed by two intersecting straight
CAB BCA ABC 180 .+ ˆ ˆ + ˆ = ° lines are at right angles to each other.

Therefore, the sum of interior angles of a Solution
triangle is 180 .° Consider the two straight lines AB and
CD intersecting at O as shown in the

Example 2�2 following figure.

Prove that the sum of two interior angles
of a triangle is equal to the exterior angle
of the third interior angle.

Solution
Consider ABC with AB extended to
D as shown in the following figure.

Required to prove that, a + b = 90°.
Construction: Draw the bisectors of the
angles formed by intersecting lines.
Required to prove that,
ˆ
CAB BCA = CBD+ ˆ ˆ . Proof
From the figure, it implies that
Proof a bb+ + a+ = 180°
From the figure, it implies that
(degree measure of a straight angle)

CAB BCA ABC 180+ ˆ ˆ + ˆ = ° Thus, 2a + 2b = 180º or
(sum of interior angles of a triangle), and 2(a + b) = 180º
ˆ
ˆ
ABC + CBD 180= ° (distributive property)
(degree measure of a straight angle), ab+ = 90°
It follows that, Therefore, the bisectors are at right
CAB BCA ABC ABC CBD+ ˆ ˆ + ˆ = ˆ + ˆ angles to each other.

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Student's Book Form Two

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