Triangles ABC and DEF are similar because they have two congruent angles and one pair of proportional sides. The value of x is 7.5.
Given:
Let's name the 2 triangles as Triangles ABC and DEF have sides of lengths 5, 10, and x+5 and 4, 10, and 5, respectively.
To prove:
Triangles ABC and DEF are similar.
Proof:
Angle-angle similarity:
Angle A and Angle D are both right angles.
Angle B and Angle E are both congruent angles, since they are opposite the congruent sides of 10.
Therefore, by the angle-angle similarity theorem, triangles ABC and DEF are similar.
Work for x:
To find x, we can use the fact that the corresponding sides of similar triangles are proportional. In this case, we can write the following proportion:
(x+5) / 5 = 10 / 4
Cross-multiplying and solving for x, we get:
4(x+5) = 50
x+5 = 12.5
x = 7.5
Therefore, x = 7.5.