Question

You can use a right triangle with leg lengths a and b and hypotenuse length c to prove the Pythagorean Theorem. Show that a² + b² = c².
Step 1: Make a square with an area of c².
b
a
Why are the corners of the unshaded shape right angles? Complete each sentence.
In each right triangle, the sum of the measures of the two acute angles is 90°
C
C
b
b
>>The two acute angles plus the corner of the unshaded shape form a straight angle that measures 180°
C
a
a
b
>>180°-90°= 90° A so each corner of the unshaded shape measures 45°

Answer 1

Complete sentences (cleaned up):

- In each right triangle, the sum of the measures of the two acute angles is 90°.
- The two acute angles from the two triangles meeting at a corner of the unshaded shape, together with that corner, form a straight angle that measures 180°.
- Therefore the corner of the unshaded shape measures 180° − 90° = 90°.
- So each corner of the unshaded shape is a right angle; hence the unshaded shape is a square with side length c and area c².

Finish the proof: Put four congruent right triangles (legs a and b, hypotenuse c) inside a big square of side a + b so they surround the central unshaded square of side c. The area of the big square is (a + b)². That area equals the area of the four triangles plus the area of the central square:

(a + b)² = 4·(1/2 ab) + c² = 2ab + c².

Expand the left side: a² + 2ab + b² = 2ab + c². Subtract 2ab from both sides to get a² + b² = c², which is the Pythagorean Theorem.