Lesson 69: Lengths of Segments

New Concept

This lesson introduces segment addition and defines two special angle pairs based on their sum.

• Segment Addition: The length of $\overline{WX}$ plus the length of $\overline{XY}$ equals the length of $\overline{WY}$.
• Complementary Angles: Complementary angles are two angles whose measures total $90^\circ$.
• Supplementary Angles: Supplementary angles are two angles whose measures total $180^\circ$.

What’s next

Property

The length of $\overline{WX}$ plus the length of $\overline{XY}$ equals the length of $\overline{WY}$.

Examples

If $\overline{LM} = 4 \text{ cm}$ and $\overline{MN} = 5 \text{ cm}$, then $\overline{LN} = 4 \text{ cm} + 5 \text{ cm} = 9 \text{ cm}$.
If $\overline{AC} = 60 \text{ mm}$ and $\overline{BC} = 26 \text{ mm}$, then $\overline{AB} = 60 \text{ mm} - 26 \text{ mm} = 34 \text{ mm}$.
If $\overline{WX} = 53 \text{ mm}$ and $\overline{XY} = 35 \text{ mm}$, then $\overline{WY} = 53 \text{ mm} + 35 \text{ mm} = 88 \text{ mm}$.

Explanation

Imagine a chocolate bar broken into two pieces. The bar's total length is simply the first piece's length plus the second. This simple rule is super useful for finding a missing segment's length if you know the total length and one of the other parts.

Property

Complementary angles are two angles whose measures total $90^\circ$.

Examples

An angle of $30^\circ$ and an angle of $60^\circ$ are complementary because $30^\circ + 60^\circ = 90^\circ$.
The complement of a $25^\circ$ angle is a $65^\circ$ angle, since $90^\circ - 25^\circ = 65^\circ$.
If $\angle A$ and $\angle B$ are complementary and $\angle A = 80^\circ$, then $\angle B = 10^\circ$.

Explanation

Think of a perfect corner, like on a book, which is a $90^\circ$ angle. If you split that corner into two smaller angles, they 'complement' each other. This means their measures add up to make the full $90^\circ$. They are the dynamic duo of right angles!

Property

Supplementary angles are two angles whose measures total $180^\circ$.

Examples

An angle of $120^\circ$ and an angle of $60^\circ$ are supplementary because $120^\circ + 60^\circ = 180^\circ$.
The supplement of a $90^\circ$ angle is another $90^\circ$ angle, since $180^\circ - 90^\circ = 90^\circ$.
If $\angle X$ and $\angle Y$ are supplementary and $\angle X = 45^\circ$, then $\angle Y = 135^\circ$.

Explanation

Picture a perfectly straight line, which forms a $180^\circ$ angle. If you draw a ray from any point on that line, you create two new angles. These two angles are 'supplementary' because they team up to perfectly form the complete straight line. They are the ultimate partners in flatness!