Prove the side of triangle is is produced then the exterior angle formed is equal to the sum of two interior opposite angle
recall that alternate interior angles between parallel lines are congruent.
Draw a line through one vertex parallel to the opposite base.
the rest is easy.
let the angles in the interior of the triangle be
x, y, and z
Extend the base at the z angle
let the exterior angle be k
we know x+y+z = 180°
we know that the z + k = 180 , (angles forming a straight line)
so x+y+z = z + k
x + y = k
state your conclusion
To prove that when a side of a triangle is produced, the exterior angle formed is equal to the sum of the two interior opposite angles, we can use the following steps:
Step 1: Draw a triangle ABC on a piece of paper or visualize it in your mind.
Step 2: Extend one side of the triangle, say side AB, to a point, which we will call D.
C
/\
/ \
side AB / \ side AC
/ \
A-------B
|
|
D
Step 3: Draw a line segment from the vertex C to the point D.
C
/\
/ \
side AB / \ side AC
/ \
A-------B
|
|
D
|
|
|
Step 4: Now, we have formed two triangles, Triangle ABD and Triangle BCD.
C
/\
/ \
side AB / \ side AC
/ \
A-------B
|
|
D
|
|
Triangle ABD | Triangle BCD
Step 5: In Triangle ABD, we have the internal angles A and B, and the external angle D.
Step 6: In Triangle BCD, we have the internal angles C and D, and the external angle BCD.
Step 7: According to the Triangle Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
Step 8: Therefore, the external angle D of Triangle ABD is equal to the sum of the internal angles C and D of Triangle BCD.
In conclusion, when a side of a triangle is produced, the exterior angle formed is equal to the sum of the two opposite interior angles.
To prove that in a triangle, if a side is produced, then the exterior angle formed is equal to the sum of the two interior opposite angles, we can use the following steps:
Step 1: Draw a triangle ABC, with side BC produced to a point D.
Step 2: Label the interior angles of the triangle as ∠A, ∠B, and ∠C, and the exterior angle formed at point D as ∠E.
Step 3: Using the property that the sum of angles in a triangle is 180 degrees, we have ∠A + ∠B + ∠C = 180 degrees.
Step 4: Since the exterior angle ∠E is formed by extending side BC, it forms a linear pair with interior angle ∠C. Therefore, the sum of ∠C and ∠E is 180 degrees.
Step 5: Now, we can prove that ∠B + ∠E = ∠C + ∠E.
Using the property of transitivity (if a = b and b = c, then a = c), we can write:
∠A + ∠B + ∠C = ∠C + ∠E (from Step 4)
∠A + ∠B = ∠E
We have already proved that ∠C + ∠E = ∠C + ∠E, so we can say:
∠A + ∠B = ∠C + ∠E = ∠C + ∠E
As a result, we can conclude that in a triangle, if a side is produced, then the exterior angle formed is equal to the sum of the two interior opposite angles.