Question

Explain why the triangles are similar and find the value of x

Answer 1

ΔABD~ΔCAD by Angle-Angle-Angle (AAA)

x = 45.3

Explanation:

∠CAD and ∠ABD are marked congruent. Since both ΔABD and ΔCAD share angle D, their third angles must both be equal to 180-∠CAD-∠D. Therefore, ΔABD~ΔCAD by Angle-Angle-Angle (AAA).

Since the triangles are similar, we can set up the following proportion:

`\displaystyle \\(x)/(40.8)=(30.2)/(27.2)`

Multiply both sides by 40.8:

`\displaystyle\\x=(30.2)/(27.2)\cdot 40.8=\boxed{45.3}`

Answer 2

45.3

Explanation:

Triangle ADB has angles BAD, D, and B.

Triangle CDA has angles C, D, and CAD.

The figure shows that angle B of triangle ADB and angle CAD of triangle CDA are congruent.

Angle D is an angle of both triangles, and it is congruent to itself.

By AA Similarity triangles ADB and CDA are similar.

This makes the ratios of the lengths of corresponding sides equal.

BD/AD = AB/AC

40.8/27.2 = x/30.2

27.2x = 40.8 × 30.2

x = 45.3