Question

45°-45°-90° triangle theorem
45°-45°-90° triangle theorem: in a 45°-45°-90° triangle, the __________ is √2 times as long as each leg.
image of triangle lmn with right angle at m, legs lm and mn labeled a, hypotenuse ln labeled c = ____√2
all 45°-45°-90° triangles are ______, because they have congruent angles. and triangles who have angles that are congruent have sides that are in ________.

Explanation:

Step1: Recall 45-45-90 Triangle Properties

In a \(45^\circ - 45^\circ - 90^\circ\) triangle (isosceles right triangle), the legs are equal (length \(a\) each), and the hypotenuse \(c\) is related to the leg by \(c = a\sqrt{2}\). So the first blank (after "the") is "hypotenuse". The formula \(c=\underline{a}\sqrt{2}\) (since \(c = a\sqrt{2}\)).

Step2: Similar Triangles Concept

Triangles with congruent angles are similar (by AA similarity criterion, as two angles \(45^\circ, 90^\circ\) are congruent). Similar triangles have sides in proportion (or "proportion" / "proportionality"). So "All \(45^\circ - 45^\circ - 90^\circ\) triangles are \(\underline{\text{similar}}\), because they have congruent angles. And triangles who have angles that are congruent have sides that are in \(\underline{\text{proportion}}\) (or "proportional" / "proportionate ratio")".

Answer:

1. (First blank: "hypotenuse"); \(c = \boldsymbol{a}\sqrt{2}\); "All \(45^\circ - 45^\circ - 90^\circ\) triangles are \(\boldsymbol{\text{similar}}\), because they have congruent angles. And triangles who have angles that are congruent have sides that are in \(\boldsymbol{\text{proportion}}\) (or "proportionality")."