Pro prove angle sum property of a triangle
Answers
Answer:
Image titled Prove the Angle Sum Property of a Triangle Step 11
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]
Image titled Prove the Angle Sum Property of a Triangle Step 2
2
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]
Image titled Prove the Angle Sum Property of a Triangle Step 3
3
State that angle PAB = angle ABC and angle CAQ = angle ACB. Because you constructed line PQ parallel to side BC of the triangle, the alternate interior angles (PAB and ABC) made by the transversal line (line AB) are congruent. Similarly, the alternate interior angles (CAQ and ACB) made by the transversal line AC are also congruent.[3]
Equation 2: angle PAB = angle ABC
Equation 3: angle CAQ = angle ACB
It is a geometric theorem that alternate interior angles of parallel lines are congruent.[4]
Image titled Prove the Angle Sum Property of a Triangle Step 4
4
Substitute angle PAB and angle CAQ in Equation 1 for angle ABC and angle ACB (as found in Equation 2 and Equation 3) respectively. Knowing that the alternate interior angles are equal lets you substitute the angles of the triangle for the angles of the line.[5]
Thus we get, Angle ABC + angle BAC + angle ACB = 180°.
In other words, in the triangle ABC, angle B + angle A + angle C = 180°. Thus, the sum of all the angles of a triangle is 180°.
Triangle is the smallest polygon which has three sides and three interior angles.
In the given triangle, ∆ABC, AB, BC, and CA represent three sides. A, B and C are the three vertices and ∠ABC, ∠BCA and ∠CAB are three interior angles of ∆ABC.
Angle Sum Property of a Triangle
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°
hence proved
