Question

Triangle XYZ is a right triangle with ZQ¯¯¯¯¯⊥XY¯¯¯¯¯¯ .

Drag and drop a correct answer into each box to complete the proof of the Pythagorean theorem.

(See images for details.)

Answer 1

1. Angle-Angle similarity Postulate 2.Proportional 3. C

Explanation:

By the Angle-Angle similarity Postulate, △YXZ~ △YZQ and △YXZ~ △ZXQ. Since similar Triangles have Proportional sides, a/c= f/a and b/c= e/b. Solving the equation for a² and b² gives a² = cf and b² =ce. Adding these together gives a²+b²=cf+ce. Factoring out the common segment gives a²+b²=c(f+e). Using the segment addition postulate gives a²+b²=c(c)

, which simplifies to a²+b²=c² (Look at image for proof)

.

Answer 2

By the Angle-Angle Similarity Postulate, △YXZ ~ △YZQ and △YXZ ~ △ZXQ. Since similar triangles have proportional sides, a/c = f/a and b/c = e/b. Solving the equation for
`a^2` and
`b^2` gives
`a^2` = cf and b^2 =ce. Adding these together gives
`a^2+b^2=cf+ce`. Factoring out the common segment gives
`a^2+b^2=c(f+e)`. Using the segment addition postulate gives
`a^2+b^2=c(c)`, which simplifies to
`a^2+b^2=c^2`.

In Mathematics and Euclidean Geometry, AAS is an abbreviation for Angle-Angle-Side and it states that when two (2) angles and the non-included side (adjacent to only one of the angles) in two triangles are all equal, then the triangles are said to be congruent.

Based on the Angle-Angle Similarity Postulate, triangle YXZ is similar to triangle YZQ and triangle YXZ is similar to triangle △ZXQ. Since all similar triangles have proportional sides, we have;

a/c = f/a and b/c = e/b.

By solving the equation for
`a^2` and
`b^2`, we have;

`a^2 = cf \\\\b^2 =ce`

By adding the above together, we have;

`a^2+b^2=cf+ce`.

By factoring out the common segment, we have;

`a^2+b^2=c(f+e)`.

By using the segment addition postulate, we have;

`a^2+b^2=c(c)\\\\a^2+b^2=c^2`