Page 171 - Maths Skills - 7

The Triangle and Its Properties 169

Fact-o-meter
� In an acute-angled triangle, all the altitudes lie inside the triangle.
� In an obtuse-angled triangle, all the altitudes donot lie inside the triangle.
� In a right-angled triangle, two altitudes coincide with two sides of a triangle.
� The point where the three altitudes of the triangle coincide is called the orthocentre of the triangle.

Let’s Attempt

Example 1: If the angles of a triangle are in the ratio 2 : 3 : 4, find the angles.
Solution: Let the measures of the given angles be 2x, 3x and 4x. Then,

2x + 3x + 4x = 180º Therefore, the measures of the three angles are
or 9x = 180º 2x = 2 × 20º = 40º, 3x = 3 × 20º = 60º,
or x = 180° = 20º and 4x = 4 × 20º = 80º.
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Example 2: In triangle ABC, the exterior angle ACD is 110º. Find the measure of ∠BAC, if ∠ABC = 54º.

Solution: We know that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.
\ ∠ACD = ∠ABC + ∠BAC A
or 110º = 54º + ∠BAC
or ∠BAC = 110º – 54º = 56º 110°

Hence, ∠BAC = 56°. B 54° C D
Example 3: Find the measures of all the angles of an equilateral triangle.

Solution: Let DABC be an equilateral triangle.
\ ∠A = ∠B = ∠C
But, we know that the sum of all the angles of a triangle is 180º.
\ ∠A + ∠B + ∠C = 180º A
or ∠A + ∠A + ∠A = 180º [Q ∠A = ∠B = ∠C]

or 3∠A = 180º

or ∠A = 180° = 60º B C
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Similarly, ∠B = ∠C = 60º
Hence, the measures of each of the angles of an equilateral triangle ∠A, ∠B and ∠C is equal to 60º.

Example 4: The vertical angle of an isosceles triangle is 64º. Find the base angles.
Solution: Let DABC be an isosceles triangle where AB = AC and base angles ∠B and ∠C are of equal
measure x. Then,
∠A + ∠B + ∠C = 180º
x + x + 64º = 180º

or 2x + 64º = 180º A
or 2x = 180º – 64º or 2x = 116º 64°
116°
or x = = 58º
2 x x
Therefore, each base angle is 58º. B C