verify the angles sum property of triangle
Answers
Answer:
To prove the angle sum property of a triangle, draw a straight-line PQ. PQ is passing through A and parallel to BC. Hence, the sum of all the interior angles of a triangle is 180° (proved).
Step-by-step explanation:
prove the angle sum property of a triangle, draw a straight-line PQ. PQ is passing through A and parallel to BC.
∠PAB + ∠BAC + ∠QAC = 180°….. (1)
Since PQ||BC and AB & AC are transversals,
∠QAC = ∠ACB (they are alternate angles)
∠PAB = ∠CBA (they are alternate angles)
Putting the value of ∠PAB and ∠QAC in equation (1), we get
∠ABC + ∠BAC + ∠ACB = 180°
Hence, the sum of all the interior angles of a triangle is 180° (proved).
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Exterior Angle Property of a Triangle
Another vital property of the triangle is the exterior angle property. One side of a triangle and the extended the adjacent side form an exterior angle. This property gives the measurement concept of exterior angles. The exterior angle property states that the exterior angle value is equal to the sum of the two opposite interior angles of the triangle. Side BC of the triangle ∆ABC is extended up to D. The exterior angle ∠ACD is equal to the sum of the two opposite interior angles ∠ABC and ∠BAC, which means ∠ABC + ∠BAC = ∠ACD.
Proof of Exterior Angle Property of a Triangle
Here, the triangle is ∆ABC. The BC side is extended up to D.
As BD is a straight line, ∠ABC and ∠ACD form an adjacent pair on BD.
So, ∠ABC + ∠ACD = 180°…… (1)
Following the angle sum property of a triangle, we can say that
∠ABC + ∠ACB + ∠BAC = 180°...…. (2)
From equation (1) & (2), we get
∠ACD = ∠ABC + ∠BAC
Hence, the exterior angle property of a triangle is proved.
Some Other Important Angle Properties of a Triangle
Besides the angle sum property and the exterior angle property of a triangle, there are some other essential properties of the angles of a triangle, and they are as follows.
The value of each angle of an equilateral triangle is 60°.
The sum of the two acute angles of a right-angled triangle is 90°.
The angle opposite to the smallest side is the smallest, and the largest angle is the opposite to the largest side.
The two angles of a triangle opposite to the two equal sides are equal.
A triangle has a maximum of one right angle or one obtuse angle.
Solved Examples
1. Find Out the Angle ∠ABC of the Triangle ∆ABC. The Exterior ∠ACD = 125° and the Other Interior Angle ∠BAC = 61°.
Ans: BC ia side of ∆ABC is extended up to D, and the exterior angle is 125°. So, the two opposite angles are ∠ABC and ∠BAC. The sum of the two angles is equal to the value of ∠ACD = 125°.
Therefore, ∠ABC = ∠ACD – ∠BAC
= 125° – 61°
= 64°
2. The Ratio of the Three Angles of a Triangle is 1:2:3. Determine the Largest Angle of the Triangle and the Type of the Triangle.
Ans: According to the angle sum property,
x + 2x + 3x = 180°
3x = 90°
Therefore, the largest angle is 90°, and it is a right-angled triangle.
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