Question

read the sentences below, filling in the blanks with items from this list. each item will be used once. side aa similarity x lengths congruent congruence sides two triangles are similar by the rule if they have two angles of the same measure. in coordinate geometry, the horizontal axis is typically labeled as the axis. similar triangles have proportional corresponding as well as congruent corresponding angles. the principle that allows the use of ratios in similar triangles to find unknown lengths is known as the proportionality. the term describes the scenario when two triangles are similar and all corresponding sides are in equal proportion. to prove triangles are similar using coordinates, one often needs to calculate the of the sides using the distance formula. when using aa similarity to prove the similarity of triangles, one must show that two angles are

Explanation:

• For the first blank, the AA (Angle - Angle) similarity rule states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
• In coordinate geometry, the horizontal axis is typically labeled as the \(x\) - axis.
• Similar triangles have proportional corresponding sides (by the definition of similar triangles).
• The principle that allows the use of ratios in similar triangles to find unknown lengths is based on the similarity of the triangles (using the properties of similar triangles).
• When two triangles are similar and all corresponding sides are in equal proportion (i.e., the ratio of corresponding sides is \(1\)), the triangles are congruent.
• To prove triangles are similar using coordinates, the lengths of the sides (calculated using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)) are often used to check for proportionality.
• When using AA similarity to prove the similarity of triangles, one must show that two angles are congruent (by the definition of AA similarity).

Answer:

1. AA similarity
2. \(x\)
3. sides
4. AA similarity (the property of similar triangles which allows ratio - based length finding is due to their similarity, and AA is a similarity rule)
5. congruence
6. lengths
7. congruent