Om is congruent to er , me is congruent to ro prove angel m is congruent to r

Answer 1

M..............|| .....................Oθ₂O'

E' θ₁E.............||.......................R

Let L denote length. I constructed a figure to satisfy the given information. (Note: Assume a line connects ME and OR. Also, assume a line connects RO' and ME'). The congruent sides must be parallel to each other by definition.

Define θ₁ = m<E'ME and θ₂ = m<O'RO

For this case, we need to prove whether θ₁ = θ2_. If true, it must hold that m<R = m<M.

Since ME is congruent to OR, L (ME) = L (OR). Similarly, L (RE) = L(OM). This implies, L (ME') = L(O'R).

But θ₂ = cos(L(O'R')/ L(OR)) = cos(L(ME')/L(ME)) = θ₁ by substitution. (*)

Therefore, m<R + θ₂ = 90--> (1): θ₂ = 90 - m<R. Similarly, (2): θ₁ = 90 - m<M.

Subtracting (1) from (2) yields θ₁ - θ₂ = -m<M + m<R. By (*), 0 = -m<M + m<R --> m<R = m< M or angle R is congruent to angle M (QED).