how can we prove that an exterior angle of a triangle has measure equal to the sum of the measures of the 2 interior angles remote from it
in triangle ABC extend BC to form the exterior angle at C, call that x
We know angle A + Angle B + angle C = 180
But angle C + x = 180, (by supplementary angle theorem)
so angle A + angle B + angle C = x + angle C
subtract angle C from both sides
angle A + angle B = x, the exterior angle
Therefore, the exterior angle of a triangle has measure equal to the sum of the measures of the 2 interior angles remote from it
rocks 12
To prove that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles remote from it, we can use the properties of triangles and angles. Here's how we can prove it step by step:
Step 1: Draw a triangle ABC. Label the vertices as A, B, and C.
Step 2: Extend one of the sides, let's say side BC, to form an exterior angle. Label the extended side as BD.
Step 3: By drawing the extended side BD, we have created two new triangles: triangle ABD and triangle BCD.
Step 4: Recall that the sum of interior angles of a triangle is always 180 degrees. Therefore, the sum of the measures of angles in triangle ABD and triangle BCD is also 180 degrees.
Step 5: In triangle ABD, the measures of angles A and B add up to 180 degrees. So, we can write it as: ∠A + ∠B = 180 degrees.
Step 6: In triangle BCD, the measures of angles B and the exterior angle at D (let's call it ∠CDE) add up to 180 degrees. Hence, we can write it as: ∠B + ∠CDE = 180 degrees.
Step 7: Since ∠A + ∠B = 180 degrees (from step 5) and ∠B + ∠CDE = 180 degrees (from step 6), we can combine the equations to get: ∠A + ∠B = ∠B + ∠CDE.
Step 8: By subtracting ∠B from both sides of the equation, we get: ∠A = ∠CDE.
Step 9: This shows that the measure of the exterior angle ∠CDE is equal to the measure of the interior angle ∠A. Thus, the measure of the exterior angle of a triangle is equal to the sum of the measures of the two interior angles remote from it.
By following these steps, we have successfully proved that an exterior angle of a triangle has a measure equal to the sum of the measures of the two interior angles remote from it.
Very brilliant answer like the way of explanation,
Thanks to help