Lesson Apps
Parallel Lines–Congruent Arcs Theorem
Press on the numbers to see the steps of the proof.
1
Draw a circle. Draw $$\mathtt{\overleftrightarrow{LP} \parallel \overleftrightarrow{AC}}$$, so that each line intersects the circle at two points.
2
Draw $$\mathtt{\overline{LC}}$$. Alternate interior angles are congruent.
$$\mathtt{m\color{orange}{\measuredangle{ACL}} = m\color{orange}{\measuredangle{CLP}}}$$
3
The Inscribed Angle Theorem tells us that $$\mathtt{m\color{orange}{\measuredangle{CLP}} = \frac{1}{2}m\color{green}{\overparen{PC}}}$$ and $$\mathtt{m\color{orange}{\measuredangle{ACL}} = \frac{1}{2}m\color{blue}{\overparen{LA}}}$$.
4
This means that $$\mathtt{\frac{1}{2}m\overparen{PC} = \frac{1}{2}m\overparen{LA}}$$.
Multiply each side by 2 to show that $$\mathtt{m\overparen{PC} = m\overparen{LA}}$$.
5
Parallel lines intersecting a circle intercept congruent arcs on the circle. The lines don't have to be horizontal.
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