构造的直觉：角的复制

Suppose that we have an angle $\angle A$. We use the construction method to create a copy $\angle B$.

We would like to prove that $\angle A\cong\angle B$. To do so, we will first walk through the construction.

First, we draw a circle $\Omega_1$ with center $A$ and some nonzero radius. Suppose that this circle meets the sides of the angle at points $X_a$ and $Y_a$. Thus, $AX_a=AY_a$.

We draw one ray with $B$ as an endpoint. Now, we draw a congruent circle $\Omega_2$ with center $B$. Denote the intersection of the circle and the ray as $X_b$.

Now, we construct a circle $\Omega_3$ with center $X_b$ and radius $X_aY_a$. Denote the intersection of the two circles as $Y_b$. Draw the ray $BY_b$, and the construction is complete.

We claim that $\angle X_aAY_a\cong\angle X_bBY_b$.

The way to go is triangle congruence!

The SSS congruence is the easiest, since we know quite a lot about side lengths.

Since $\Omega_1\cong\Omega_2$, and since they are circles, we know the following: $$AX_a=AY_a=BX_b=BY_b$$ Moreover, since the circle $\Omega_3$ has center $X_b$ and radius $X_aY_a$, $$X_aY_a=X_bY_b$$ By SSS congruence, we know the following: $$\triangle AX_aY_a\cong\triangle BX_bY_b$$ Thus, since corresponding parts of congruent triangles are congruent (CPCTC), we know: $$\angle X_aAY_a\cong\angle X_bBY_b$$ Indeed, these are our two angles $\angle A$ and $\angle B$, and we have shown that they are congruent.

Thus, the proof is complete, and this is a valid construction for copying an angle.

To copy an angle:

1. Draw 2 intersecting lines, $l$ and $m$. Let $B$ be their intersection point.

2. Draw another line, $p$, away from $l$ and $m$. Mark a point on $p$, $Y$. $Y$ will be the vertex of the congruent angle you will form.

3. Draw a circle of any radius with $B$ as its center so that it marks two points, $A$ and $C$, on lines $l$ and $m$, respectively.

4. Now draw a congruent circle to the one you just drew with $Y$ as the center. Label the intersection point of $p$ and $\circ Y$ $X$.

5. Draw a circle with center $A$ and radius $AC$.

6. Draw a congruent circle to the one you just drew with $X$ as the center.

7. Draw a line from $Y$ to the intersection point of $\circ Y$ and $\circ X$. Label the intersection point $Z$. So, now you have that $\angle ABC \cong \angle XYZ$. Congratulations, you have just constructed congruent angles!