Question

unit 8: 30-60-90 special right triangles
do not: answer the following question in complete sentences.
what do you notice about these answers?
(three right triangle diagrams with pythagorean theorem calculations: $2^2+b^2=4^2$, $4+b^2=16$, $b^2=12$, $b=2\sqrt{3}$; $3^2+b^2=6^2$, $9+b^2=36$, $b^2=27$, $b=3\sqrt{3}$; $5^2+b^2=10^2$, $25+b^2=100$, $b^2=75$, $b=5\sqrt{3}$)

1. what do you notice about the angles of each triangle?

what i notice about the angles of each triangle is that ______

1. what do you notice about the sides of each triangle?

what i notice about the sides of each triangle is that ______
what if you cut an equilateral triangle in half?

• get 2 ______

when do we use the 30-60-90 rule?
the ______ is used to find the missing side
of a ______ that has angle measurements

Explanation:

1. For the angles: Each triangle is a right triangle with a 30° angle, a 60° angle, and a 90° angle, so they are all 30-60-90 special right triangles.
2. For the sides: The shortest leg (opposite 30°) is half the length of the hypotenuse, and the longer leg (opposite 60°) is $\sqrt{3}$ times the shortest leg. This matches the calculated values: $2\sqrt{3} = 2 \times \sqrt{3}$, $3\sqrt{3} = 3 \times \sqrt{3}$, $5\sqrt{3} = 5 \times \sqrt{3}$, with hypotenuses 4, 6, 10 (double the shortest legs 2, 3, 5).
3. Cutting an equilateral triangle in half creates 2 congruent 30-60-90 right triangles.
4. The 30-60-90 rule is used for right triangles with those angle measurements.

Answer:

1. What I notice about the angles of each triangle is that they are all 30-60-90 right triangles (one right angle, one 30°, one 60°).
2. What I notice about the sides of each triangle is that the hypotenuse is twice the shorter leg, and the longer leg is $\sqrt{3}$ times the shorter leg.
3. Get 2 congruent 30-60-90 right triangles.
4. The 30-60-90 rule is used to find the missing side of a right triangle that has angle measurements of 30°, 60°, and 90°.