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.

\(\mathtt{m\color{orange}{\measuredangle{ACL}} = m\color{orange}{\measuredangle{CLP}}}\)

\(\mathtt{m\color{orange}{\measuredangle{ACL}} = m\color{orange}{\measuredangle{CLP}}}\)

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 \(\mathtt{m\overparen{PC} = m\overparen{LA}}\).

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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