Question

1. match.

the 45°-45°-90° triangle theorem states that in a 45°-45°-90° triangle, these legs are
and the length of the hypotenuse is
times the length of a leg.

a. $\sqrt{2}$
b. congruent
c. specific
d. $\sqrt{3}$

1. match.

the 30°-60°-90° triangle theorem states that in a 30°-60°-90° triangle, the length of the
hypotenuse is
the length of the shorter leg, and the length of
the longer leg is
times the length of the shorter leg.

a. 3 times
b. $\sqrt{3}$
c. twice
d. $\sqrt{2}$

Explanation:

1)

Step1: Analyze 45-45-90 triangle legs

In a 45°-45°-90° triangle, the two legs are congruent (since the two non - right angles are equal, by the isosceles triangle theorem, the sides opposite them are equal). So the first blank should be filled with "congruent" (option b).

Step2: Analyze 45-45-90 triangle hypotenuse

For a 45°-45°-90° triangle, if the length of each leg is \( l \), using the Pythagorean theorem \( c^{2}=l^{2} + l^{2}=2l^{2}\), then \( c = \sqrt{2}l \). So the length of the hypotenuse is \( \sqrt{2} \) times the length of a leg (option a).

Step1: Analyze 30-60-90 triangle hypotenuse

In a 30°-60°-90° triangle, the side opposite the 30° angle is the shorter leg. Let the length of the shorter leg be \( x \). The hypotenuse is twice the length of the shorter leg (this is a well - known property of 30 - 60 - 90 triangles). So the first blank should be filled with "twice" (option c).

Step2: Analyze 30-60-90 triangle longer leg

The length of the longer leg (opposite the 60° angle) can be found using the Pythagorean theorem or the property of 30 - 60 - 90 triangles. If the shorter leg is \( x \) and the hypotenuse is \( 2x \), then the longer leg \( y \) satisfies \( y^{2}+x^{2}=(2x)^{2}\), \( y^{2}=4x^{2}-x^{2}=3x^{2}\), so \( y = \sqrt{3}x \). So the length of the longer leg is \( \sqrt{3} \) times the length of the shorter leg (option b).

Answer:

First blank: b. congruent; Second blank: a. \( \sqrt{2} \)

2)