Midsegment and Triangle Proportionality Theorems Cheat Sheet

Key Facts

• If $M$ and $N$ are midpoints of $AB$ and $AC$ in $\triangle ABC$, then $MN \parallel BC$ and $MN = \frac{1}{2}BC$.
• If $MN$ is a midsegment of $\triangle ABC$, then the third side has length $BC = 2MN$.
• If $DE \parallel BC$ in $\triangle ABC$ with $D$ on $AB$ and $E$ on $AC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.
• If $DE \parallel BC$ in $\triangle ABC$, then the smaller triangle is similar to the larger triangle, so $\triangle ADE \sim \triangle ABC$.
• For similar triangles $\triangle ADE \sim \triangle ABC$, corresponding side ratios are equal, so $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$.
• The converse of the Triangle Proportionality Theorem says that if $\frac{AD}{DB} = \frac{AE}{EC}$, then $DE \parallel BC$.
• When a line parallel to one side of a triangle cuts the other two sides, always match corresponding segments in the same order before writing a proportion.
• A midpoint divides a segment into two equal parts, so if $M$ is the midpoint of $AB$, then $AM = MB$ and $AB = 2AM$.

Vocabulary

Midsegment

A midsegment is a segment that connects the midpoints of two sides of a triangle.

Midpoint

A midpoint is a point that divides a segment into two congruent parts.

Parallel Lines

Parallel lines are coplanar lines that never intersect and have the same direction.

Proportion

A proportion is an equation showing that two ratios are equal, such as $\frac{a}{b} = \frac{c}{d}$.

Similar Triangles

Similar triangles have congruent corresponding angles and proportional corresponding side lengths.

Converse

A converse reverses the hypothesis and conclusion of a theorem to create a related statement.