Question

model it

complete the problems about right triangles.

1 look at the triangles below.

a. write the missing angle measure(s). then circle all the right triangles.
b. there are special names for the sides of a right triangle. the side opposite the right angle is the hypotenuse. the two sides that meet to form the right angle are called the legs. label each side of any right triangles above as either hypotenuse or leg.
2 there is an important relationship between the lengths of the sides of a right triangle: for a right triangle with leg lengths a and b and hypotenuse length c, (a^{2}+b^{2}=c^{2}). this relationship is called the pythagorean theorem.
a. use the information above to label the length of each side of the right triangle with a, b, or c.
b. write hypotenuse or legs in each blank to help explain what (a^{2}+b^{2}=c^{2}) means in words:

Explanation:

Step1: Find missing angles

The sum of angles in a triangle is 180°. For the first triangle with a 90° angle, if we assume the other two angles are equal (isosceles right - triangle), each of the non - right angles is $\frac{180 - 90}{2}=45^{\circ}$. The second triangle has angles 60°, 60°, and $180-(60 + 60)=60^{\circ}$, so it is not a right - triangle. The third triangle has angles 30°, 60°, and 90°. The right - triangles are the first and the third.

Step2: Label sides of right - triangles

In the first right - triangle, the side opposite the 90° angle is the hypotenuse, and the other two equal sides are the legs. In the third right - triangle, the side opposite the 90° angle is the hypotenuse, and the other two sides are the legs.

Step3: Label sides for Pythagorean Theorem

In a right - triangle, label the two legs as $a$ and $b$ and the hypotenuse as $c$. The legs are the sides that form the right angle, and the hypotenuse is the side opposite the right angle.

Step4: Explain Pythagorean Theorem in words

The sum of the squares of the lengths of the legs ($a^{2}+b^{2}$) is equal to the square of the length of the hypotenuse ($c^{2}$).

Answer:

a. First triangle: Missing angles are 45°, 45°. Third triangle: Missing angle is 30°. Right - triangles are the first and the third.
b. In right - triangles, label sides as described.
c. In a right - triangle, label the legs as $a$ and $b$ and the hypotenuse as $c$.
d. The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.