Module 5 Study Guide: Similarity and Triangle Properties

Vocabulary and core theorem review flashcards based on the Module 5 Study Guide covering triangle similarity postulates, proportional reasoning, midsegments, and practical applications.

Ratio

A mathematical relationship or comparison between two numbers or quantities expressed by division.

Proportion

An equation demonstrating that two ratios or fractions are equivalent, frequently written in the form $\frac{a}{b} = \frac{c}{d}$.

Similar Figures

Geometric figures that share identical shape but differ in size, possessing congruent corresponding interior angles and proportional corresponding side lengths.

Triangle Midsegment Theorem

A theorem stating that a segment connecting the midpoints of two sides of a triangle runs parallel to the third side and equals half the length of that parallel third side, expressed as $\text{Midsegment} = \frac{1}{2}(\text{Parallel Side})$.

Angle-Angle (AA) Similarity Postulate

A postulate establishing that if two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar.

Side-Side-Side (SSS) Similarity Theorem

A theorem establishing that if the corresponding side lengths of two triangles are in proportional ratio to one another, the triangles are similar.

Side-Angle-Side (SAS) Similarity

A similarity criterion stating that if two sides of one triangle are proportional to two corresponding sides of another triangle and their included angles are congruent, the triangles are similar.

Side-Splitter Theorem

A theorem stating that if a line segment is parallel to one side of a triangle and intersects the remaining two sides, it divides those two sides proportionally.

Triangle-Angle Bisector Theorem

A theorem stating that an angle bisector within a triangle divides the opposite side into segments proportional to the lengths of the other two adjacent sides.

Scale Factor

The fixed ratio of any two corresponding linear dimensions of two similar geometric figures.

Corresponding Sides Ratio

The principle that in similar triangles such as $\Delta ABC \sim \Delta DEF$, corresponding sides form equivalent ratios: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$.

Perimeter Ratio of Similar Triangles

The theorem establishing that the ratio of the perimeters of two similar triangles equals the ratio of their corresponding side lengths: $\frac{\text{Perimeter}_1}{\text{Perimeter}_2} = \frac{s_1}{s_2}$.

Indirect Measurement (Shadow Reckoning)

A method using proportional ratios of similar right triangles created by sunlight and shadows to determine unknown vertical heights, such as setting $\frac{\text{Tower Height}}{\text{Tower Shadow}} = \frac{\text{Post Height}}{\text{Post Shadow}}$.

Midsegment Value Relation (Study Guide Problem 7)

The algebraic application where midsegment $CD = x + 4$ is set equal to half of parallel side $AB = 4x - 5$, solving to $x = 6.5$, $CD = 10.5$, and $AB = 21$.

Tower Height Calculation (Study Guide Problem 11)

The proportion solving for tower height $h$ from a $65\text{-foot}$ ($780\text{ in}$) shadow and a $42\text{-inch}$ post casting a $9\text{-inch}$ shadow: $\frac{h}{780} = \frac{42}{9}$, resulting in $h = 3640\text{ in}$ or $303\frac{1}{3}\text{ ft}$.