Question

state and prove ASP of triangle ​

Answer 1

ℏḙʏᾰ Պᾰтḙ. ..

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Figure 1 Triangle ABC

Theorem 1: Angle sum property of triangle states that the sum of interior angles of a triangle is 180°.

Proof: Consider a ∆ABC, as shown in the figure below. To prove the above property of triangles, draw a line PQ←→ parallel to the side BC of the given triangle.

Proof for Angle Sum Property of a Triangle

Since PQ is a straight line, it can be concluded that:

∠PAB + ∠BAC + ∠QAC = 180° ………(1)

SincePQ||BC and AB, AC are transversals,

Therefore, ∠QAC = ∠ACB (a pair of alternate angle)

Also, ∠PAB = ∠CBA (a pair of alternate angle)

Substituting the value of ∠QAC and∠PAB in equation (1),

∠ACB + ∠BAC + ∠CBA= 180°

Thus, the sum of the interior angles of a triangle is 180°.

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ℏ✺℘ḙ тℏ!ṧ ℏḙℓ℘ṧ

Answer 2

Draw a line parallel to side BC of the triangle that passes through the vertex A. Label the line PQ. Construct this line parallel to the bottom of the triangle.[1]


Write the equation angle PAB + angle BAC + angle CAQ = 180 degrees. Remember, all of the angles that comprise a straight line must be equal to 180°. Because angle PAB, angle BAC, and angle CAQ combine together to make line PQ, their angles must sum to 180°. Call this Equation 1.[2]